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MACMII.LAX AND CO., LIMITED 

LONDON . ROMHAY . C.MXIII A 
MELBOURNE 

THK MA< "Ml LI. AN COMPANY 

Ni:\V YORK . BOSTON . CHICAGO 

DALLAS . <AN KKANCISCO 

THE MACMILLAN CO. OF CANADA, LTD. 

TOR' 



Plate I, 




GR,KAT STAR-CLOUD IN SAGITTARIUS. 



STELLAR MOVEMENTS 

AND THE STRUCTURE 

OF THE UNIVERSE 



BY 

A. S. EDDINGTON 

'\ 

M.A. (CANTAB.), M.Sc. (MANCHESTER), B.Sc. (Lo.\n.), F.R.S. 

Plnmian Professor of Astronomy, University of Cambridge 



MACMILLAN AND CO., LIMITED 
ST. MARTIN'S STREET, LONDON 

1914 












COPYR1GH7* 



PREFACE 

THE purpose of this monograph is to give an account of 
the present state of our knowledge of the structure of the 
sidereal universe. This branch of astronomy has become 
especially prominent during the last ten years ; and many 
new facts have recently been brought to light. There is 
every reason to hope^that the next few years will be equally 
fruitful ; and it may seem hazardous at the present stage to 
attempt a general discussion of our knowledge. Yet perhaps 
at a time like the present, when investigations are being 
actively prosecuted, a survey of the advance made may be 
especially helpful. 

The knowledge that progress will inevitably lead to a 
readjustment of ideas must instil a writer with caution; 
but I believe that excessive caution is not to be desired. 
There can be no harm in building hypotheses, and weaving 
explanations which seem best fitted to our present partial 
knowledge. These are not idle speculations if they help 
us, even temporarily, to grasp the relations of scattered 
fjicts, and to organise our knowledge. 

No attempt has been made to treat the subject histori- 
cally. I have preferred to describe the results of inves- 



300340 



vi PREFACE 

tigations founded on the most recent data rather than early 
pioneer researches. One inevitable result I particularly 
regret ; many of the workers who have prepared the way 
for recent progress receive but scanty mention. Sir W. 
Herschel, Kobold, Seeliger, Newcomb and others would 
have figured far more prominently in a historical account. 
But it was outside my purpose to describe the steps by 
which knowledge has advanced ; it is the present situation 
that is here surveyed. 

So far as practicable I have endeavoured to write for 
the general scientific reader. It was impossible, without 
too great a sacrifice, to avoid mathematical arguments 
altogether ; but the greater part of the mathematical 
analysis has been segregated into two chapters (VII and 
X). Its occasional intrusion into the remaining chapters 
will, it is hoped, not interfere with the readable character 
of the book. 

I am indebted to Prof. G. E. Hale for permission to re- 
produce the two photographs of nebulae (Plate 4), taken by 
Mr. G. W. Ritchey at the Mount Wilson Observatory, and 
to the Astronomer Royal, Dr. F. W. Dyson, for the three re- 
maining plates taken from the Franklin-Adams Chart of the 
Sky. This represents but a small part of my obligation to 
I )r. Dyson ; at one time and another nearly all the subjects 
treated in this book have been discussed between us, and I 
make no attempt to discriminate the idcns wliir-h I owe 
to him. There are many other astronomers from whose 
conversation, consciously or unconsciously, I have drawn 
material for this work. 



PREFACE vii 

I have to thank Mr. , P. J. Melotte of the Royal 
Observatory, Greenwich, who kindly prepared the three 
Franklin- Adams photographs for reproduction. 

Dr. S. Chapman, Chief Assistant at the Royal Observa- 
tory, Greenwich, has kindly read the proof-sheets, and I am 
grateful to him for his careful scrutiny and advice. 

I also desire to record my great indebtedness to the 
Editor of this series of monographs, Prof. R. A. Gregory, 
for many valuable suggestions and for his assistance in 
passing this work through the press. 

A. S. EDDINGTON. 

THE OBSERVATORY, CAMBRIDGE. 
April, 1914. 



CONTENTS 

CHAPTER I 



PAOF. 

THE DATA OF OBSERVATION 1 



CHAPTER II 

GENERAL OUTLINE . 30 

CHAPTER III 

THE NEAREST STARS 40 

CHAPTER IV 

-MOVING CLUSTERS 54 

CHAPTER V 

THE SOLAR MOTION 71 

CHAPTER VI 

THE TWO STAR STREAM- 86 

CHAPTER VII 

THE TWO STAR STREAMS MATHEMATICAL THEORY 127 

CHAPTER VIII 

PHENOMENA ASSOCIATED WITH SPECTRAL TYPE 154 

CHAPTER IX 

COUNTS OF STARS 184 



x CONTENTS 

CHAPTER X 

GENERAL STATISTICAL INVESTIGATIONS 201 

CHAPTER XI 

THE MILKY WAY, STAR-CLUSTERS AND NEBULAE 232 

CHAPTER XII 

DYNAMICS OF THE STELLAR SYSTEM 246 

INDEX 263 



LIST OF ILLUSTRATIONS 



PLATES 

/ 

PLATE I. The Great Star-cloud in Sagittarius Frontispiece 

,, II. Nebulous dark space in Corona Austrina facing paye 236 
,, III. The Greater Magellanic Cloud ,, 240 

(Nebula, Canes Venatici. N.G.C. 5194-5 \ 249 

' iNebula, Coma Berenices. H.V. 24 / 



FIGURES 



PAGI 



1. Hypothetical Section of the Stellar System 31 

2. Moving Cluster in Taurus. (Boss.) 57 

3. Geometrical Diagram 59 

4. Moving Cluster of " Orion "Stars in Perseus 64 

5. Simple Drift Curves 88 

6. Observed Distribution of Proper Motions. (Groombridge Cata- 

logue R. A. 14*> to 18 h , Dec. +38 to +70 : .) .... 89 

7. Calculated Distributions of Proper Motions 92 

8. Observed and Calculated Distributions of Proper Motions. 

(Boss, Region VIII.) 93 

9. Diagrams for the Proper Motions of Boss's " Preliminary 

General Catalogue " 95, 96, 97, 98 

10. Convergence of the Directions of Drift I. from the 17 Regions . 99 

11. Convergence of the Directions of Drift II. from the 17 Regions . 99 

12. Geometrical Diagram 101 

13. Distribution of Large Proper Motions (Dyson) .... 105 

14. Mean Proper Motion Curves 112 

xi 



xii LIST OF ILLUSTRATIONS 

Hi;. PAGE 

15. Mean Proper Motion Curve (Region of Boss's Catalogue) . . 116 

16. Comparison of Two-drift and Three-drift Theories . . .122 

17. Distribution of Drifts along the Equator (Halm) .... 123 

18. Comparison of Two-drift and Ellipsoidal Hypotheses . . . 135 

19. Absolute Magnitudes of Stars (Russell) 171 

20. Number of Stars brighter than each Magnitude for Eight Zones 

(Chapman and Melotte) 188 

21. Curves of Equal Frequency of Velocity 220 

22. Geometrical Diagram 249 



STELLAR MOVEMENTS AND THE 
STRUCTURE OF THE UNIVERSE 

CHAPTER I 

THE DATA OF OBSERVATION 

IT is estimated that the number of stars which could 
be revealed by the most powerful telescopes now in use 
amounts to some hundreds of millions. One of the princi- 
pal aims of stellar astronomy is to ascertain the relations 
and associations which exist among this multitude of 
individuals, and to study the nature and organisation of 
the great system which they constitute. This study is as 
yet in its infancy ; and, when we consider the magnitude 
of the problem, we shall scarcely expect that progress 
towards a full understanding of the nature of the sidereal 
universe will be rapid. But active research in this branch 
of astronomy, especially during the last ten years, has led 
to many results, which appear to be well established. It 
has become possible to form an idea of the general dis- 
tribution of the stars through space, and the general 
character of their motions. Though gaps remain in our 
knowledge, and some of the most vital questions are as 
yet without an answer, investigation along many different 
lines has elicited striking facts that may well be set down. 
It is our task in the following pages to coordinate these 
results, and review the advance that has been made. 

B 



MOVEMENTS CHAP. 

Until recent years the study of the bodies of the solar 
system formed by far the largest division of astronomy ; 
but with that branch of the subject we have nothing to do 
here. From our point of view the whole solar system is 
only a unit among myriads of similar units. The system 
of the stars is on a scale a million-fold greater than that of 
the planets ; and stellar distances exceed a million times 
the distances of the comparatively well-known bodies 
which circulate round the Sun. Further, although we 
have taken the stars for our subject, not all branches of 
stellar astronomy fall within the scope of this book. It 
is to the relations between the stars to the stars as a 
society that attention is here directed. We are not con- 
cerned with the individual peculiarities of stars, except in 
so far as they assist in a broad classification according to 
brightness, stage of development, and other properties. 
Accordingly, it is not proposed to enter here into the 
more detailed study of the physical characters of stars ; 
and the many interesting phenomena of Variable Stars 
and Novae, of binary systems, of stellar chemistry and 
temperatures are foreign to our present aim. 

The principal astronomical observations, on which the 
whole superstructure of fact or hypothesis must be based, 
may be briefly enumerated. The data about a star which 
are useful for our investigations are : 

1. Apparent position in the sky. 

2. Magnitude. 

3. Type of spectrum, or Colour. 

4. Parallax. 

5. Proper motion. 

6. Radial velocity. 

In addition, it is possible in some rather rare cases to find 
the mass or density of a star. This is a matter of import- 
ance, for presumably one star can only influence another 
by means of its gravitational attraction, which depends 
on the masses. 



1 THE DATA OF OBSERVATION 3 

This nearly exhausts the list of characteristics which 
are useful in investigating the general problems of stellar 
distribution.* It is only in the rarest cases that the com- 
plete knowledge of a star, indicated under the foregoing 
six heads, is obtainable ; and the indirect nature of most of 
the processes of investigation that are adopted is due to 
the necessity of making as much use as possible of the very 
partial knowledge that we do possess. 

The observations enumerated already will now be con- 
sidered in order. The apparent position in the sky needs 
no comment ; it can always be stated with all the accuracy 
requisite. 

Magnitude. The magnitude of a star is a measure of 
its apparent brightness ; unless the distance is known, 
we are not in a position to calculate the intrinsic or 
absolute brightness. Magnitudes of stars are measured 
on a logarithmic scale. Starting from a sixth magnitude 
star, which represents an arbitrary standard of brightness 
of traditional origin, but now fixed with sufficient pre- 
cision, a star of magnitude 5 is one from which we receive 

2 '5 12 times more light. Similarly, each step of one magni- 
tude downwards or upwards represents an increase or 
decrease of light in the ratio 1 : 2*5 12. f The number is so 
chosen that a difference of five magnitudes corresponds to 
a light ratio of 100 = (2*5 12) 5 . The general formula is 

= - -- 



where 

.Lj, L 2 are the intensities of the light from two stars 
mj, m 2 are their magnitudes. 

Magnitude classifications are of two kinds : photometric 
(or visual), and photographic ; for it is often found that of 

* We should perhaps add that the separations and periods of binary stars 
are also likely to prove useful data. 

f It is necessary to warn the reader that there are magnitude systems still 
in common use that of the Bonn Durchmusterung, that used by many 
double-star observers, and even the magnitudes of Boss's Preliminary 
General Catalogue (1910) which do not conform to this scale. 

B 2 



4 STELLAR MOVEMENTS CHAP. 

two stars the one which appears the brighter to the eye 
leaves a fainter image on a photographic plate. Neither of 
the two systems has been defined very rigorously; for, when 
stars are of different colours, a certain amount of personality 
exists in judging equality of light by the eye, and, if a 
photographic plate is used, differences may arise depending 
on the colour-sensitiveness of the particular kind of plate, 
or on the chromatic correction of the telescope object- 
glass. As the accuracy of magnitude-determinations im- 
proves, it will probably become necessary to adopt more 
precise definitions of the visual and photographic scales ; 
but at present there appears to be no serious want of 
uniformity from this cause. But the distinction between 
the photometric and photographic magnitudes is very 
important, and the differences are large. The bluer the 
colour of a star, the greater is its relative effect on the 
photographic plate. A blue star and a red star of the 
same visual brightness may differ photographically by as 
much as two magnitudes. 

The use of a logarithmic scale for measuring brightness 
possesses many advantages ; but it is liable to give a 
misleading impression of the real significance of the 
numbers thus employed. It is not always realised how 
very rough are the usual measures of stellar brightness. 
If the magnitude of an individual star is not more than 
IU T in error, we are generally well satisfied ; yet this 
means an error of nearly 10 per cent, in the light-intensity. 
Interpreted in that way it seems a rather poor result. 
An important part of stellar investigation is concerned 
with counts of the number of stars within certain limits of 
magnitude. As the number of stars increases about 
three-fold for each successive step of one magnitude, it is 
clear that all such work will depend very vitally on the 
absence of systematic error in the adopted scale of 
magnitudes; an error of two- or three- tenths of a magni- 
tude would affect the figures profoundly. The establish- 



i THE DATA OF OBSERVATION 5 

ment of an accurate magnitude-system, with sequences of 
standard stars, has been a matter of great difficulty, and 
it is not certain that even now a sufficiently definitive 
system has been reached. The stars which coine under 
notice range over more than twenty magnitudes, corre- 
sponding to a light ratio of 100,000,000 to 1. To 
sub-divide such a range without serious cumulative error 
would be a task of great difficulty in any kind of physical 
measurement. 

The extensive researches of the Harvard Observatory, 
covering both hemispheres of the sky, are the main basis 
of modern standard magnitudes. The Harvard sequence 
of standard stars in the^neighbourhood of the North Pole, 
extending by convenient steps as far as magnitude 21, 
at the limit reached with the 60-inch reflector of the 
Mount Wilson Observatory, now provides a suitable scale 
from which differential measures can be made. The 
absolute scale of the Harvard sequence of photographic 
magnitudes has been independently tested by F. H. Scares, 
at Mount Wilson, and S. Chapman and P. J. Melotte, at 
Greenwich. Both agree that from the tenth to the 
fifteenth magnitudes the scale is sensibly correct. But 
according to Scares a correction is needed between the 
second and ninth magnitudes, l m '00 on the Harvard scale 
being equivalent to l m *07 absolute. If this result is 
correct, the error introduced into statistical discussions 
must be quite appreciable. 

For statistical purposes there are now available deter- 
minations of the magnitudes of the stars in bulk made at 
Harvard, Potsdam, Gottingen, Greenwich and Yerkes 
Observatories. The revised Harvard photometry gives the 
visual magnitudes of all stars down to about 6 m '5 ; the 
Potsdam magnitudes, also visual, carry us in a more limited 
part of the sky as far as magnitude 7 m '5. The Gottingen 
determinations, which are absolute determinations, inde- 
pendent of but agreeing very fairly with the Harvard 



6 STELLAR MOVEMENTS CHAP. 

sequences, provide photographic magnitudes over a large 
zone of the sky for stars brighter than 7 m< 5. The Yerkes 
investigation gives visual and photographic magnitudes 
of the stars within 17 of the North Pole down to 7 m> 5. 
A series of investigations at Greenwich, based on the 
Harvard sequences, provides statistics for the fainter stars 
extending as far as magnitude 17, and is a specially 
valuable source for the study of the remoter parts of the 
stellar system. 

So far we have been considering the apparent brightness 
of stars and not their intrinsic brightness. The latter 
quantity can be calculated when the distance of the star 
is known. We shall measure the absolute luminosity in 
terms of the Sun as unit. The brightness of the Sun has 
been measured in stellar magnitudes and may be taken to 
be 26 m 'l, that is to say it is 26*1 magnitudes brighter 
than a star of zero magnitude.* From this the luminosity 
L of a star, the magnitude of which is m, and parallax 
is &", is given by 

Iog 10 L = 0'2 - 0'4 x w -2 log lo Z. 

The absolute magnitudes of stars differ nearly as widely 
as their apparent magnitudes. The feeblest star known is 
the companion to Groombridge 34, which is eight magni- 
tudes fainter than the Sun. Estimates of the luminosities 
of the brightest stars are usually very uncertain ; but, to 
take only results which have been definitely ascertained, 
the Cepheid Variables are on an average seven magnitudes 
brighter than the Sun. Probably this luminosity is 
exceeded by many of the Orion type stars. There is thus 
a range of at least fifteen magnitudes in the intrinsic 
brightness, or a light ratio of 1,000,000 to 1. 

* The most recent researches give a value -26*5 (Ceraski, Annul* of the 
Obwvatory of Moscow, 1911). It is best, however, to regard the unit of 
luminosity as a conventional unit, roughly representing the Sun. ami defined 
by the formula, rather than to keep changing tin- measures of stellar 
luminosity every time a better determination of the Sun's stellar magnitude 
is mad*-. 



i THE DATA OF OBSERVATION 7 

Type of Spectrum. For the type of spectrum 
various systems of classification have been used by astro- 
physicists, but that of the Draper Catalogue of Harvard 
Observatory is the most extensively employed in work on 
stellar distribution. This is largely due to the very 
complete classification of the brighter stars that has been 
made on this system. The classes in the supposed order 
of evolution are denoted by the letters : 

O, B, A, F, G, K, M, N. 

A continuous gradation is recognised from to M, and a 
more minute sub-division is obtained by supposing the 
transition from one class to the next to be divided into 
tenths. Thus B5A, usually abbreviated to B5, indicates a 
type midway between B and A ; G2 denotes a type 
between Gr and K, but more closely allied to the former. 
It is usual to class as Type A all stars from AO to A9 ; but 
presumably it would be preferable to group together the 
stars from B6 to A5, and this principle has occasionally 
been adopted. 

For our purposes it is not generally necessary to consider 
what physical peculiarities in the stars are represented by 
these letters, as the knowledge is not necessary for discussing 
the relations of motion and distribution. All that we require 
is a means of dividing the stars according to the stage of 
evolution they have attained, and of grouping the stars 
with certain common characteristics. It may, however, be 
of interest to describe briefly the principles which govern 
the classification, and to indicate the leading types of 
spectrum. 

Tracing an imaginary star, as it passes through the 
successive stages of evolution from the earliest to the 
latest, the changes in its spectrum are supposed to pursue 
the following course. At first the spectrum consists 
wholly of diffuse bright bands on a faint continuous back- 
ground. The bands become fewer and narrower, and faint 



8 STELLAR MOVEMENTS CHAP. 

absorption lines make their appearance ; the first lines seen 
are those of the various helium series, the well- 
known Balmer hydrogen series, and the " additional " 
or "sharp" hydrogen series.* The last is a spectrum 
which had been recognised in the stars by E. C. Pickering 
in 1896 but was first produced artificially by A. Fowler 
in 1913. The bright bands now disappear, and in the 
remaining stages the spectrum consists wholly of absorp- 
tion lines and bands, except in abnormal individual 
stars, which exceptionally show bright lines. The next 
phase is an enormous increase in the intensity of the 
true hydrogen spectrum, the lines becoming very wide 
and diffused ; the other lines disappear. The lines H 
and K of calcium and other solar lines next become 
evident and gain in intensity. After this the hydrogen 
lines decline, though long remaining the leading feature of 
the spectrum ; first a stage is reached in which the calcium 
spectrum becomes very intense, and afterwards all the 
multitudinous liues of the solar spectrum are seen. After 
passing the stage reached by our Sun, the chief feature is 
a shortening of the spectrum from the ultraviolet end, a 
further fading of the hydrogen lines, an increase in the 
number of fine absorption lines, and finally the appearance 
of bands due to metallic compounds, particularly the flutings 
of titanium oxide. The whole spectrum ultimately approxi- 
mates towards that of sunspots. 

Guided by these principles, we distinguish eight leading 
types, between which, however, there is a continuous series 
of gradations. 

TYPE (WOLF-RAYET TYPE) The spectrum consists of 
bright bands on a faint continuous background ; of these 
the most conspicuous have their centres at \\ 4633, 
4651, 4686, 5693, and 5813. The type is divided into 

* The practical worU of A. Fowler and the theoretical researches of N. I5 tin- 
leave little doubt that this spectrum is due to helium, notwithstanding its 
simple numerical relation to the hydrogen spectrum. 



i THE DATA OF OBSERVATION 9 

five divisions, Oa, Ob, Oe, Od, and Oe, marked by varying 
intensities and widths of the bands. Further, in Od and 
Oe dark lines, chiefly belonging to the helium and helium- 
hydrogen series, make their appearance. 

TYPE B (ORION TYPE) This is often called the helium 
type owing to the prominence of the lines of that 
element. In addition there are some characteristic lines 
the origin of which is unknown, as well as both the " sharp " 
and Balmer series. The bright bands seen in Type are no 
longer present; in fact, they disappear as early as the 
sub-division OeoB, which is therefore usually reckoned the 
starting point of the Orion type. 

TYPE A (SiRiAN TYPE). The Balmer series of hydro- 
gen is present in great intensity, and is far the most 
conspicuous feature of the spectrum. Other lines are 
present, but they are relatively faint. 

TYPE F (CALCIUM TYPE). The hydrogen series is still 
very conspicuous, but not so strong as in the preceding 
type. The narrow H and K lines of calcium have become 
very prominent, and characterise this spectrum. 

TYPE G (SOLAR TYPE). The Sun may be regarded as 
a typical star of this class, the numerous metallic lines 
having made their appearance. 

TYPE K. The spectrum is somewhat similar to the 
last. It is mainly distinguished by the fact that the 
hydrogen lines, which are still fairly strong in the G stars, 
are now fainter than some of the metallic lines. 

TYPE M. The spectrum is now marked by the appear- 
ance of flutings, due to titanium oxide. It is remarkable 
that the spectrum should be dominated so completely by 
this one substance. Two successive stages are recognised, 
indicated by the sub-divisions Ma and Mb. The long- 
period variable stars, which show bright hydrogen lines in 
addition to the ordinary characteristics of Type M, form 
the class Md. 

TYPE N. The regular progression of the types 



io STELLAR MOVEMENTS CHAP. 

terminates with Mb. There is no transition to Type N, 
and the relation of this to the foregoing types is uncertain. 
It is marked by characteristic flu tings attributed to com- 
pounds of carbon. The stars of both the M and N types 
are of a strongly reddish tinge. 

It is sometimes convenient to use the rather less detailed 
classification of A. Secchi. Strictly speaking his system 
relates to the visual spectrum, and the Draper notation to 
the photographic ; but the two can well be harmonised. 

Secchi's Type I. includes Draper B and A 

II. F, G, andK 

III. M 

TV N 

>j -"-' -^ 

As comparatively few of the stars in any catalogue belong 
to the last two types, this classification is practically a 
separation into two groups, which are of about equal size. 
This is a very useful division when the material is too 
scanty to admit of a more extended discussion. 

From time to time there are indications that the Draper 
classification has not succeeded in separating the stars into 
really homogeneous groups. According to Sir Norman 
Lockyer, there are stars of ascending temperature and of 
descending temperature in practically every group ; so 
that, for example, the stars enumerated under K are a 
mixture of two classes, one in a very early, the other in a late, 
stage of evolution. In the case of Type B, H. Ludendorff 1 
has found considerable systematic differences in the 
measured radial motions of the stars classed by Lockyer as 
ascending and descending respectively (pointing, however, 
to real differences not of motion but of physical state, 
which have introduced an error into the spectroscopic 
measurements). E. Hertzsprung 2 has pointed out that 
the presence or absence of the c character on Miss Maury's 
classification (i.e., sharply defined absorption-lines) corre- 
sponds to an important difference in the intrinsic 
luminosities of the stars. Hitherto, however, it has 



THE DATA OF OBSERVATION 



ii 



been usual to accept the Draper classification as at least 
the most complete available for our investigations. 

Colour-Index. Stars may be classified according to 
colour as an alternative to spectral type. Both methods 
involve dividing the stars according to the nature of 
the light emitted by them, and thus have something in 
common. Perhaps we might not expect a very close 
correspondence between the two classifications ; for, whilst 
colour depends mainly on the continuous background of 
the spectrum, the spectral type is determined by the fine 
lines and bands, which can have little direct effect on the 
colour. Nevertheless a close correlation is found between 
the two characters, owing no doubt to the fact that both 
are intimately connected with the effective temperature of 
the star. 

The most convenient measure of colour is afforded by 
the difference, photographic minus visual magnitude ; this 
is called the colour-index. The relation between the 
spectral type and the colour-index is shown below. 





Colour index according to 


Spectral Type. 










King. 


Schwarzschild. 




m 


m 


Bo 


-0-31 


-0-64 


Ao 


o-oo 


-0-32 


Fo 


+ 0-32 


o-oo 


Go 


+ 0-71 


+ 0-32 


Ko 


+ 1-17 


+ 0-95 


M 


+ 1-68 


+ 1-89 



I 

King's results 3 refer to the Harvard visual and photographic 
scales ; Schwarzschild's 4 to the Gottingen photographic and 
Potsdam visual determinations. Allowing for the constant 
difference, depending on the particular type of spectrum 
for which the photographic and visual magnitudes are 



12 STELLAR MOVEMENTS CHAP. 

made to agree, the two investigations confirm one another 
closely. 

The foregoing results are derived from the means of a 
considerable number of stars, but the Table may be 
applied to individual stars with considerable accuracy. 
Thus the spectral type can be found when the colour- 
index is known, and conversely. At least in the case of 
the early type stars, the spectral type fixes the colour- 
index with an average uncertainty of not more than O m 'l ; 
for Types G and K larger deviations are found, but the 
correlation is still a very close one. 

Yet another method of classifying stars according to the 
character of the light emitted is afforded by measures of 
the "effective wave-length.'* If a coarse grating, consist- 
ing of parallel strips or wires equally spaced, is placed in 
front of the object-glass of a telescope, diffraction images 
appear on either side of the principal image. These 
diffraction images are strictly spectra, and the spot which 
a measurer would select as the centre of the image will 
depend on the distribution of intensity in the spectrum. 
Each star will thus have a certain effective wave-length 
which will be an index of its colour, or rather of the 
appreciation of its colour by the photographic plate. For 
the same telescope and grating the interval between the two 
first diffracted images is a constant multiple of the effective 
wave-length. The method was first used by K. Schwarz- 
schild in 1895 ; and an important application of it was 
made by Prosper Henry to determine the effect of 
atmospheric dispersion on the places of the planet 
Eros. It has been applied to stellar classification by 
E. Hertzsprung. 5 

Parallax. The annual motion of the earth around 
the Sun causes a minute change in the direction in which 
a star is seen, so that the star appears to describe a 
small ellipse in the sky. This periodic displacement is 
superposed on the uniform proper- motion of the star, 



i THE DATA OF OBSERVATION 13 

which is generally much greater in amount ; there is, 
however, no difficulty in disentangling the two kinds of 
displacement. Since we are only concerned with the 
direction of the line joining the two bodies, the effect of 
the earth's motion is the same as if the earth remained at 
rest, and the star described an orbit in space equal to that 
of the earth, but with the displacement reversed, so that 
the star in its imaginary orbit is six months ahead of the 
earth. This orbit is nearly circular ; but, as it is generally 
viewed at an angle, it appears as an ellipse in the sky. 
In any case the major axis of the ellipse is equal to the 
diameter of the earth's orbit ; and, since the latter length 
is known, a determination of its apparent or angular 
magnitude affords a means of calculating the star's 
distance. The parallax is defined as the angle subtended 
by one astronomical unit (the radius of the earth's orbit) 
at the distance of the star, and is equivalent to the major 
semiaxis of the ellipse which the star appears to describe. 

The measurement of this small ellipse is always made 
relatively to some surrounding stars, for it is hopeless to 
make absolute determinations of direction with the 
necessary accuracy. The relative parallax which is thus 
obtained needs to be corrected by the amount of the 
average parallax of the comparison stars in order to obtain 
the absolute parallax. This correction can only be 
guessed from our general knowledge of the distances of 
stars similar in magnitude and proper motion to the 
comparison stars ; but as it could seldom be more than 
0"*01, not much uncertainty is introduced into the final 
result from this cause. 

The parallax is the most difficult to determine of all the 
quantities which we wish to know, and for only a very 
few of the stars has it been measured with any approach 
to certainty. Until some great advance is made in means 
of measurement, all but a few hundreds of the nearest 
stars must be out of range of the method. But so 



i 4 STELLAR MOVEMENTS CHAP. 

Laborious are the observations required, that even these 
will occupy investigators for a long while. In general, the 
published lists of parallaxes contain many that are ex- 
tremely uncertain, and some that are altogether spurious. 
Statistical investigations based on these are liable to be 
very misleading. Nevertheless, it is believed that by 
rejecting unsparingly all determinations but those of the 
highest refinement, some important information can be 
obtained, and in Chapter III. these results are discussed. 
In addition, determinations which are not individually of 
high accuracy may be used for finding the mean parallaxes 
of stars of different orders of magnitude and proper motion, 
provided they are sensibly free from systematic error ; 
these at least serve to check the results found by less 
direct methods. 

The measurements are generally made either by photo- 
graphy or visually with a heliometer. The former method 
now appears to give the best results owing to the greater 
focal-length of the instruments available. It has also the 
advantage of using a greater number of comparison stars, 
so that there is less chance of the correction to reduce to 
absolute parallax being inaccurate. Some early determin- 
ations with the heliometer are, however, still unsurpassed. 
The meridian-circle is also used for this work, and consider- 
able improvement is shown in the more recent results of 
this method ; but we still think that meridian parallaxes 
are to be regarded with suspicion, and have deemed it 
best not to use them at all in Chapter III. 

A convenient unit for measuring stellar distances is the 
parsec* or distance which corresponds to a parallax of one 

* The parsec, a p<>i tin.-uitrau-namr sujj^i-strd by H. H. Turner, will be 
used throughout this book. Several different units of stellar distance have, 
however, been employed by investigators. Kobold's Sternweite is identical 
with the parsec. Seeliger's Siriusu'eite corresponds to a parallax 0"'2, and 
( 'h;irli-r's Xiriometer to a million astronomical units or purallax 0"'206. The 
light-year which, notwithstanding its inconvenience and irrelevance, has 
sometimes crept from popular use into technical investigations, is e<]ual to 
0'31 parsecs. 



i THE DATA OF OBSERVATION 15 

second of arc. This is equal to 206,000 astronomical units 
or about 19,000,000,000,000 miles. The nearest fixed 
star, a Centauri, is at a distance of 1*3 parsecs. There 
are perhaps thirty or forty stars within a distance of five 
parsecs, and, of course, the number at greater distances will 
increase as the cube of the limiting distance, so long as 
the distribution is uniform. But these nearest stars are 
not by any means the brightest visible to us ; the range 
in intrinsic luminosity is so great that the apparent magni- 
tude is very little clue to the distance. A sphere of radius 
thirty parsecs would probably contain 6000 stars ; but 
the 6000 stars visible to the average eye are spread through 
a far larger volume of space. It appears indeed that some 
of the naked -eye stars are situated in the remotest parts 
of the stellar system. 

A parallax-determination may be considered first-class 
if its probable error is as low as 0"'01. If, for instance, 
the measured parallax is 0"'050"*01, it is an even chance 
that the true value lies between 0"'06 and 0"'04, and we 
probably should not place much confidence in any nearer 
limits than 0"'07 and 0"'03. This is equivalent to saying 
that the star is distant something between fourteen and 
thirty-three parsecs from us. It will be seen that, when 
the parallax is as low as 0"'05, even the best measures give 
only the very roughest idea of the distance of the star, 
and for smaller values the information becomes still more 
vague. Clearly, to be of value a parallax must be at least 
0"'05. It may be estimated that there are not more than 
2000 stars so near as this, and a very large proportion of 
these will be fainter than the tenth magnitude. The chances 
are that, of five plates of the international Carte du Ciel 
taken at random, only one will be fortunate enough to 
pick up a serviceable parallax, and even that is likely to be 
a very inconspicuous star, which would evade any but the 
most thorough search. The prospect of so overwhelming a 
proportion of negative results suggests that, for the present 



1 6 STELLAR MOVEMENTS CHAP. 

at any rate, work can be most usefully done on special 
objects for which an exceptionally large proper motion 
affords an a priori expectation of a sensible parallax. A 
star of parallax 0"'05 may be expected to have a proper 
motion of 20" per century or more, and that seems to be 
a reasonable limit to work down to. 

It will generally be admitted that a most valuable 
extension of our knowledge will result from precise 
measures of the distances of as many as possible of the 
individual stars that come within the range above 
mentioned. But many investigators have also sought to 
determine the mean parallaxes of stars of different 
magnitudes or motions. When the individual distances 
are too uncertain, the means of a large number may still 
have some significance. Whilst some useful results can be 
and have been obtained by this kind of research, its 
possibilities seem to be very limited. Generally speaking, 
this class of determination requires even greater refine- 
ment than the measurement of individual parallaxes ; 
refinement which is scarcely yet within reach. For example, 
the mean parallax of stars of the sixth magnitude is 
0"*014 (perhaps a rather high estimate) ; that of the 
comparison stars would probably be about half this, so that 
the relative parallax actually measured would be 0"'007. 
The possible systematic errors depending on magnitude 
and colour (the mean colour of the sixth magnitude is 
perhaps different from the ninth) make the problem of 
determining this difference one of far higher difficulty than 
that of measuring the parallax of an individual star. It 
means gaining almost another decimal place beyond the 
point yet reached. We need not dwell on the enormous 
labour of observing the necessary fifty or one hundred 
sixth magnitude stars to obtain this mean with reason- 
able accuracy ; it might well be thought worth the 
trouble ; but there is no evidence that systematic 
errors have as yet been brought as low as 0" - 001 



i THE DATA OF OBSERVATION 17 

even in the best work, and indeed it seems almost 
inconceivable. 

From these considerations it appears that parallax- 
determinations should be directed towards : 
t (l) Individual stars with proper motions exceeding 20" 
a century. This will yield many negative results, but a 
fair proportion of successes. 

(2) Classes of stars with proper motions less than 20", 
but still much above the average. These parallaxes will 
have to be found individually, but for the most part only 
the mean result for a class will be of use. 

(3) A possible extension to classes of stars not dis- 
tinguished by large proper motion, provided it is realised 
that a far higher standard is required for this work, and 
that a freedom from systematic error as great as 0"*001 can 
be ensured. 

Proper Motion. --For stellar investigation the 
proper motions, i.e.. the apparent angular motions of the 
stars, form most valuable material. For extension in our 
knowledge of magnitudes, parallaxes, radial velocities, and 
spectral classification, we shall ultimately come to depend 
on improved equipment and methods of observation ; but 
the mere lapse of time enables the proper motions to 
become known with greater and greater accuracy, and the 
only limit to our knowledge is the labour that can be 
devoted, and the number of centuries we are content to 
wait. 

Proper motions of stars differ greatly in amount, but in 
general the motion of any reasonably bright star (e.g., 
brighter than 7 m '0) is large enough to be detected in the 
time over which observations have already accumulated. 
Whilst it is quite the exception for a star to have a 
measurable parallax, it is exceptional for the proper motion 
to be insensible. It may be useful to give some idea of 
the certainty and trustworthiness of the proper motions in 
ordinary use, though the figures are^ necessarily only 

C 



1 8 STELLAR MOVEMENTS CHAP. 

approximate. When the probable error is about 1" per 
century in both coordinates, the motion may be considered 
to be determined fairly satisfactorily ; the Groombridge and 
Carrington Catalogues, largely used in statistical investiga- 
tions, are of about this order of accuracy. A higher 
standard probable error about 0" 6 per century is 
reached in Boss's "Preliminary General Catalogue of 6188 
Stars," which is much the best source of proper motions 
at present available. For some of the fundamental stars 
regularly observed at a large number of places throughout 
the last century the accidental error is as low as 0*'2 per 
century ; but the inevitable systematic errors may well 
make the true error somewhat larger. Various sources of 
systematic error, particularly uncertainties in the constant 
of precession and the motion of the equinox, may render 
the motions in any region of the sky as much as 0"*5 per 
century in error ; it is unlikely that the systematic error 
of the best proper motions can be greater than this, except 
in one or two special regions of southern declination, 
where exceptional uncertainty exists. 

"We may thus regard the proper motions used in 
statistical researches as known with a probable error of 
not more than 1" per century in right ascension and 
declination. Koughly speaking, an average motion is from 
3" to 7" per century. A centennial motion of more than 
20" is considered large, although there are some stars which 
greatly exceed this speed. The fastest of all is C.Z. 
5 h 243 a star of the ninth magnitude found by J. C. 
Kapteyn and R. T. A. Innes on the plates of the Cape 
Photographic Durchmusterung which moves at the rate 
of 870" per century. This speed would carry it over an 
arc equal to the length of Orion's belt in just above a 
thousand years. Table 1 shows the stars at present 
known of which the centennial speed exceeds 300". The 
number of faint stars on this list is very striking ; and, as 
our information practically stops at the ninth magnitude, it 



THE DATA OF OBSERVATION 



may be conjectured that there are a number of still fainter 
stars yet to be detected. 

TABLE 1. 
Stars with large Proper Motion. 



Name. j& 


Dec. nnual 

1900. PP er 
Motion. 


Magnitude. 


h. m. 
C.Z. 5 h 243 5 8 
Groombridge 1830 ... 11 47 
Lacaille 9352 . . . 22 59 




-45-0 870 
+ 38-4 7-07 
-36'4 7-02 


8-3 
6-5 

7-4 


Cordoba 32416 
61 1 Cygni 21 2 


-37-8 6-07 
+ 38-3 5-25 


8-5 
5*6 


Lalonde 21185 . 10 58 


+ 36 '6 477 


7 '6 


f Indi . . . . 21 56 


-57 -2 4-67 


47 


Lalonde 21258 . 11 


+ 44-0 4-46 


8-9 


o 2 Eridani 4 11 


- 7'8 4-08 


4-5 


fO.A. (s.) 14318 15 5 
\O.A. (s.) 14320. 15 5 


-16-0 3-76 
-15-9 376 


9-6 
9'2 


p. Cassiopeiae . . 12 


+ 54-4 375 


5'3 


a 1 Centauri 14 33 


-60-4 3*66 


0*3 


Lacaille 8760 ... . 21 11 
e Eridani .... 3 16 


-39-2 3-53 
-43'4 3-15 


7-3 
4 '3 


O.A. (x.) 11677 .... 11 15 


+ 66-4 3-03. 


9-2 



Comparatively little is known of the motions of stars 
fainter than the ninth magnitude. The Carrington proper 
motions, discussed by F. W. Dyson, carry us down to 
10 m '3 for the region within 9 of the North Pole. A 
number of the larger proper motions of faint stars in the 
Oxford Zone of the Carte du Ciel have been published by 
H. H. Turner and F. A. Bellamy. 6 Further, by the reduc- 
tion of micrometric measurements,^ G. C. Comstock 7 has 
obtained a number of proper motions extending even to 
the thirteenth magnitude. There is no difficulty to be 
expected in securing data for faint stars ; but the work has 
been taken up comparatively recently, and the one essential 
is lapse of a sufficient time. 

Radial Velocity. The velocity in the line of sight 
is measured by means of a spectroscope. In accordance 
with Doppler's Principle, the lines in the spectrum 
of a star are displaced towards the red or the violet 

c 2 



20 STELLAR MOVEMENTS CHAP. 

(relatively to a terrestrial comparison spectrum) according 
as the star is receding from or approaching the earth. 
Unlike the proper motion, the radial motion is found 
directly in kilometres per second, so that the actual linear 
speed, unmixed with the doubtful element of distance, is 
known. Hitherto it has scarcely been possible to measure 
the velocities of stars fainter than the fifth magnitude, but 
that limitation is now being removed. The main difficulty 
in regard to the use of the results is the large proportion 
of spectroscopic binary stars, about one in three or four of 
the total number observed. As the orbital motion is often 
very much larger than the true radial velocity, it is 
essential to allow sufficient time to elapse to detect any 
variation in the motion, before assuming that the measures 
give the real secular motion, of which we are in search. 
Another uncertainty arises from possible systematic errors 
affecting all the stars belonging to a particular type of 
spectrum. There is reason to believe that the measured 
velocity of recession of the Type B stars is systematically 
5 km. per sec. too great. 8 This may be due to errors in 
the standard wave-lengths employed, or to a pressure-shift 
of the lines under the physical conditions prevailing in 
this kind of star. Smaller errors affect the stars of other 
types. 

Apart from possible systematic error, a remarkable 
accuracy has been attained in these observations. For a 
star with sharp spectral lines a probable error of under 
0'25 km. per sec. is well within reach. Stars of Types 
B and A have more diffuse lines and the results are not 
quite so good ; but the accuracy even in these cases is far 
beyond the requirements of the statistician. The observed 
velocities range up to above one hundred km. per sec., but 
speeds greater than sixty km. per sec. are not very common. 
The greatest speed yet measured is that of Lalande 19 GO, 
viz. , 325 km. per sec. The next highest is C.Z. 5 h 243, already 
mentioned as having the greatest apparent motion acr<> 



i THE DATA OF OBSERVATION 21 

the sky ; it is observed to be receding at the rate of 
242 km. per sec., or 225 km. per sec. if we make 
allowance for the Sun's own motion. As these figures 
refer to only one component of motion, the total speeds of 
stars are sometimes considerably greater. 

Radial motions of about 1400 stars have now been 
published, the great bulk of the observations having been 
made at the Lick Observatory. Most of this material 
only became accessible to investigators in 1913, and 
there has scarcely yet been time to make full use of the 
new data. 

There are a few systems which can be observed both as 
visual and as spectroscopic binaries. In such cases it is 
possible to deduce the distance of the star by a method 
quite independent of the usual parallax determinations. 
From the visual observations, the period and the other 
elements of the orbit can be found. The dimensions, 
however, are all expressed in arc, i.e., in linear measure 
divided by the unknown distance of the star. From these 
elements we can calculate for any date the relative velocity 
in the line of sight of the two components ; but this also 
will be expressed as a linear velocity divided by the 
unknown distance. By comparing this result with the 
same relative velocity measured spectrographically, and 
therefore directly in linear measure, the distance of the 
system can be derived. This method is of very limited 
application ; but in the case of a Centauri it has given a 
very valuable confirmation of the parallax determined in 
the ordinary way. It increases our confidence that the 
usual method of measuring stellar distances is a sound one. 

Mass and Density. Knowledge of the masses 
and densities of stars is derived entirely from binary 
systems. The sources of information are of three kinds : 

(1) From visual binaries. 

(2) From ordinary spectroscopic binaries. 

(3) From eclipsing variables. 



22 



STELLAR MOVEMENTS 



CHAP. 



The combined mass of the two components of a binary 
system can be found from the length of the major semi- 
axis of the orbit a and the period P by the formula 



Here the masses are expressed in terms of the Sun's mass 
as unit, and the astronomical unit and the year are taken 
as the units of length and time. 

In a well-observed visual orbit, all the elements are 
known (except for an ambiguity of sign of the inclination), 
but the major axis is expressed in arc. This can be 
converted into linear measure, if the parallax has been 
determined ; and hence m^ + m 2 can be found. When further, 
besides the relative orbit, a rough absolute orbit of one of 
the components has been found, by meridian observations 
or otherwise, the ratio f m l \ i m. 2 is determinate, and m 1 and 
m 2 are deduced separately. Owing to the difficulty of 
determining parallaxes, cases of a complete solution of this 
kind are rare. They are, however, sufficient to indicate 
the fact that the range in the masses of the stars is not at 
all proportionate to the huge range in their luminosities. 

TABLE 2. 

Well-detei-mined Masses of Stars. 





Combined System 


Brighter Com- 
ponent 


Star 






Mass 
(Sun=l) 


Period 
Years 


a 


Parallax 


Luminos- 
ity 
(Sun-1) 


Spectral 
Type 


( Herculis . . 


1-8 


::i D 1-35 


0-14 


5-0 


G 


Procyon . . . 


1-3 


39-0 


4-05 


0-32 


97 


F5 


Sirius .... 


3"4 


48-8 


7-59 


0-38 


48-0 


A 


a Centauri . . 


1-9 


81-2 1771 


0-76 


2-0 


G 


7'i uphiuchi . 


2-5 


88-4 


4-55 


0-17 


1-1 


K 


o 2 Eridani . 


n-7 


lHO-n 479 


0-17 


0-84 


G 


rj Cassiopeiae * . 


J-0 


328-0 9-48 


0-20 


1-4 


F5 












8 



* Another published orbit P = 508y -12'2" gives the mass =0'9. The 
great uncertainty of the orbit appears to have little effect on the result. 



i THE DATA OF OBSERVATION 23 

Table 2 contains all the systems, of which the masses can be 
ascertained with reasonable accuracy, i.e., systems for which 
good orbits 9 and good parallax-determinations 10 have been 
published. Possibly some of the more doubtful orbits 
would have been good enough for the purpose, but I doubt 
if the list could be much extended without lowering the 
standard. 

Another fact which appears is that the ratio of the 
masses of the two components of a binary is generally not 
far from equality, notwithstanding considerable differences 
in the luminosity. Thus Lewis Boss 11 in ten systems found 
that the ratio of the mass of the faint star to the brighter 
star ranged from 0'33 to IT,* the mean being 071. The 
result is confirmed by observations of such spectroscopic 
binaries as show the lines of both components, though in 
this case the disparity of luminosity cannot be so great. 

Even w r hen the parallax is not known, important 
information as to the density can be obtained. Consider 
for simplicity a system in which one component is of 
negligible mass ; the application to the more general case 
requires only slight modifications, provided mjm. 2 is known 
or can be assumed to have its average value. 
Let 

d be the distance of the star 

6 its radius 

S its surface brightness 

L, I its intrinsic and apparent luminosities 

M its mass 

p its density 

y the constant of gravitation 

Then 



L = nV 

and 



* Excluding one very doubtful result. 



24 STELLAR MOVEMENTS CHAP. 

From these 



- 7 is the semi-axis of the orbit in arc, and I and P are 
a 

observed quantities ; consequently the coefficient of S% is 
known. We have thus an expression for the density in 
terms of the surface brightness, and can at least compare 
the densities of those stars which, on spectroscopic evi- 
dence, may be presumed to have similar surface conditions. 

The deusity is found to have a large range, many of the 
stars being apparently in a very diffused state with 
densities perhaps not greater than that of atmospheric air. 

The spectroscopic binaries also give some information as 
to the masses of stars. The formula (m x + m 2 ) = a 3 /P 2 is 
applicable, and as a is now found in linear measure it is 
not necessary to know the parallax. The quantity deduced 
from the observations is, however, in this case a sin i, 
where i is the inclination of the plane of the orbit. The 
inclination remains unknown, except when the star is an 
eclipsing variable * or in the rare case when the system 
is at the same time a visual and a spectroscopic binary. 
For statistical purposes, such as comparing the masses of 
different types of stars, we may assume that in the mean 
of a sufficient number of cases the planes of the orbits will 
be distributed at random, and can adopt a mean value for 
sin i. Thus from spectroscopic binaries the average masses 
of classes, but not of individual stars, can be found. 

In the case of eclipsing variable stars, the densities of 
the two components can be deduced entirely from the light- 
curve of the star. Although these are necessarily spectro- 
scopic binary systems, observations of the radial velocity 
are not needed, and are not used in the results. The 
actual procedure, which is due to H. N. Russell and 
H. Shapley, 12 is too complicated to be detailed here, as 

* In this case it is evident that * must be nearly 90, and accordingly sin i 
may be taken as unity. 



i THE DATA OF OBSERVATION 25 

it bears only incidentally on our subject ; but the general 
principle may be briefly indicated, it will be easily 
realised that the proportionate duration of the eclipse and 
other features of the light curve do not depend on the 
absolute dimensions of the system, but on the ratio of the 
three linear quantities involved, viz., the diameters of the 
two stars and the distance between their centres. By 
strictly geometrical considerations therefore, we find the 
radii r lt r 2 of the stars expressed in terms of the unknown 
semi-axis of the orbit a as unit. Now the relation between 
the mass and density of a star involves the cube of the 
radius, and the dynamical relation between the mass and 
the period involves the cube of a. Thus, on division, the 
absolute masses and the unknown unit a disappear simul- 
taneously, and we are left with the density expressed in 

7* /T 

terms of the period and the known ratios , . Tiie 

a a 

key to the solution is that in astronomical units the 
Dimensions of density are (time)" 2 ; the density thus 
depends on the period, and on the ratios, but not the 
absolute values, of the other constants of the system. 

The densities found in this way are not quite rigorously 
determined. It is necessary to assume a value of the 
ratio of the masses of the two stars ; as already explained, 
this ratio does not differ widely from unity, but in extreme 
cases the results may be as much as fifty per cent, in error 
from this cause. Further, the darkening at the limb of 
the star has some effect on the determination, and the 
assumed law of darkening is hypothetical. By taking 
different assumptions, between which the truth is bound 
to lie, it can be shown that these uncertainties do not 
amount to anything important, when regard is had to the 
great range in stellar densities which is actually found. 

We may conclude this account of the nature of the 
observations, on which our knowledge of the stellar 



26 STELLAR MOVEMENTS CHAP. 

universe is based, by a reference to J. C. Kapteyn's " Plan 
of Selected Areas." When the study of stars was confined 
mainly to those brighter than the seventh magnitude, and 
again when it was extended as far as the ninth, or tenth, 
a complete survey of all the stars was not an impossible 
aim, and indeed all the data obtainable could well be 
utilised. But investigations are now being pushed towards 
the fifteenth and even lower magnitudes. These fainter 
stars are so numerous that it is impossible and unneces- 
sary to do more than make a selection. As the different 
kinds of observation for parallax, proper motion, magnitude, 
spectral type, and radial velocity are highly specialised and 
usually carried out at different observatories, some co-opera- 
tion is necessary in order that so far as possible the 
observations may be concentrated on the same groups of 
stars. Kapteyn's 13 plan of devoting attention to 206 
selected areas, distributed all over the sky, so as to 
cover all varieties of stellar distribution, has met with 
very general support. The areas have their centres on or 
near the circles of declination, 0, 15, 30, 45, 
=t 60, 75, 90. The exact centres have been chosen 
with regard to various practical considerations ; but the 
distribution is very nearly uniform. In addition to the 
main " Plan," 46 areas in the Milky Way have been 
chosen, typical of its main varieties of structure. The 
area proper consists of a square 75' x 75', or alternatively 
a circle of 42' radius ; but the dimensions may be extended 
or diminished for investigations of particular data. 

The whole scheme of work includes nine main sub- 
divisions : (1) A Durchmusterung of the areas. (2) 
Standard photographic magnitudes. (3) Visual and 
photovisual magnitudes. (4) Parallaxes. (5) Differential 
proper motions. (6) Standard proper motions. (7) Spectra. 
(8) Radial velocities. (9) Intensity of the background of 
the sky. The Durchmusterung is well advanced ; it will 
include all stars to 17 m , the positions being given with a 



i THE DATA OF OBSERVATION 27 

probable error of about 1" in each co-ordinate, and the 
magnitudes (differential so far as this part of the work is 
concerned) with a probable error of O m *l. Considerable 
progress has been made with the determination of sequences 
of standard photographic magnitudes for each area. The 
work of determining the visual magnitudes has been partly 
accomplished for the northern zones. For parallaxes, most 
of the areas have been portioned out between different 
Observatories ; the greatest progress has been made at the 
Cape Observatory for the southern sky, but the plates 
have not yet been measured. From what has already 
been stated with regard to the practical possibilities of 
parallax-determinations, it will be seen that there is some 
doubt as to the utility of this part of the Plan. For the 
proper motions of faint stars, the work has necessarily been 
confined mainly to obtaining plates for the initial epoch. At 
the Radcliffe Observatory, 150 plates have been stored away 
undeveloped, ready for a second exposure after a suitable 
interval, but in most cases it is intended to rely on the 
parallax plates for giving the initial positions. For standard 
proper motions in the northern sky, observations are 
shortly to be started at Bonn ; these will serve for com- 
parison with older catalogues, but they may also be regarded 
as initial observations for more accurate determinations 
in the future. Determinations of spectral type as far as 
the ninth magnitude, made at Harvard, will shortly be 
available for these areas and, indeed, for the whole sky. 
The extension to the eleventh magnitude is very desirable, 
and is one of the most urgent problems of the whole 
Plan. Radial velocity determinations are being pressed 
as far as 8 m *0 at Mount Wilson, but rapid progress 
is not to be expected until the completion of the 100- 
inch reflector. A valuable, though unofficial, addition 
to the programme is E. A. Fath's Durchmusterung u of 
all the nebulae in the areas from the North pole to 
Dec. -15. 



28 STELLAR MOVEMENTS CHAP. 

REFERENCES. CHAPTER I. 

1. Ludendorff, Astr. Nach., No. 4547. 

2. Hertzsprung, Astr. Nach., No. 4296. 

3. King, Harvard Annalx, Vol. 59, p. 152. 

4. Schwarzschild, " Aktinometrie, " Teil B, p. 19. 

5. Hertzsprung, Potsdam Publications, No. 63 ; Astr. Nach., No. 4362 
(contains a bibliography). 

6. Monthly Notices, Vol. 74, p. 26. 

7. Comstock, Pub. Washburn Observatory, Vol. 12, Pt. 1. 

8. Campbell, Lick Bulletin, No. 195, p. 104. 

9. Aitken, Lick Bulletin, No. 84. 

10. Kapteyn and Weersma, Groningen Publications, No. 24. 

11. Boss, Preliminary General Catalogue of 6188 Stars, Introduction, 

p. 23. 

12. Russell and Shapley, Astrophysical Journal, Vol. 35, p. 315, et seq. 

13. Kapteyn, "Plan of Selected Areas"; ditto, ''First and Second 

Reports " ; Monthly Notices, Vol. 74, p. 348. 

14. Fath, Astronomical Journal, Nos. 658-9. 

BIBLIOGRAPHY. 

Magnitudes. The Harvard Standard Polar Sequence is given in Harvard 
Circular, No. 170. For an examination by Scares, see Astrophysical Journal, 
Vol. 38, p. 241. 

The chief catalogues of visual stellar magnitudes are : 
** Revised Photometry," Harv. Ann., Vols. 50 and 54. 
Miiller and Kempf, Potsdam Publications, Vol. 17. 

For photographic magnitudes : 

Schwarzschild, "Aktinometrie," Teil B (Gottingen Abhandlungen, 

Vol. 8, No. 4). 

Greenurich Astrographic Catalogue, Vol. 3 (advance section). 
Parkhurst, Astrophysical Journal, Vol. 36, p. 169. 

A useful discussion of the methods of determining photographic magni- 
tude will be found in an R.A.S. " Council Note," Monthly Notices, 
Vol. 73, p. 291 (1913). 

Spectral Type*. The most extensive determinations are to be found in 
Jltirv. Ann., Vol. 50, which gives the type for stars brighter than 6 m> 5. 
Scattered determinations of many fainter stars also exist. It is understood 
that a very comprehensive catalogue containing the types of 200,000 stars 
will shortly be issued from Harvard. 

A description of the principles of the Draper classification is given in 
Harv. Ann., Vol. 28, pp. 140, 146. 

Parallaxes. The principal sources are : 

Peter, Abhandlunyen kb'niglichsiichsische Gesell. der Wissenschaften, 

Vol. 22, p. 239, and Vol. 24, p. 179. 
Gill, Cape Annals, Vol. 8, Pt. 2 (1900). 



i THE DATA OF OBSERVATION 29 

Schlesinger, Aatrophyaical Journal, Vol. 34, p. 28. 
Russell and Hinks, Astronomical Journal, No. 618-9. 
Elkin, Chase and Smith, Yale Transactions, Vol. 2, p. 389. 
Slocum and Mitchell, J-sfro^/j^/m^ Jmrrwd, Vol.38, p. 1. 

Very useful compilations of the parallaxes taken from all available sources 
are given by Kapteyn and Weersma, Qroningen Publications, No. 24, and 
by Bigourdan, Bulletin Axtronmiiiiiue, Vol. 26. Fuller references are given 
in these publications. 

Proper Motions. Lewis Boss's Preliminary General Catalogue of 6188 
Stars, which includes proper motions of all the brighter stars, supersedes 
many earlier collections. 

Dyson and Thackeray's New Reduction of Groombridge's Catalogue contains 
4243 stars, including many fainter than the eighth magnitude, within 
50' of the North Pole. 

Still fainter stars are contained in the Greenwich-Carrington proper 
motions discussed by Dyson. Some of these are published in the Second 
Nine-Year Catalogue (1900), but certain additional corrections are required 
to the motions there given (see Monthly Notices, Vol. 73, p. 336). The 
complete list (unpublished) contains 3735 stars. 

Porter's Catalogue, Cincinnati Publications. No. 12, gives 1340 stars 
of especially large proper motion. 

Radial Velocities. Catalogues containing a practically complete summary 
of the radial velocities at present available for discussion are given by 
Campbell in Lick Bulletin, Nos. 195, 211, and 229. These contain about 
1350 stars. 



CHAPTER II 

GENERAL OUTLINE 

THIS chapter will be devoted to a general description of 
the sidereal universe as it is revealed by modern researches. 
The evidence for the statements now made will appear 
gradually in the subsequent part of the book, and minor 
details will be filled in. But it seems necessary to 
presume a general acquaintance with the whole field of 
knowledge before starting on any one line of detailed 
investigation. At first sight it might seem possible to 
divide the subject into compartments the distribution of 
the stars through space, their luminosities, their motions, 
and the characters of the different spectral types but it is 
not possible to pursue these different branches of inquiry 
independently. Any one mode of investigation leads, as a 
rule, to results in which all these matters are involved 
together, and no one inquiry can be worked out to a 
conclusion without frequent reference to parallel investiga- 
tions. We have therefore adopted the unusual course of 
placing what may be regarded as a summary thus early in 
the book. 

In presenting a summary, we may claim the privilege of 
neglecting many awkward difficulties and uncertainties 
that arise, promising to deal fairly with them later. We 
can pass over alternative explanations, which for the 
moment are out of favour ; though they need to be kept 



30 



CH. ii GENERAL OUTLINE 31 

alive, for at any moment new facts may be found, which 
will cause us to turn to them again. The bare outline, 
devoid of the details, must not be taken as an adequate 
presentation of our knowledge, and in particular it will 
fail to convey the real complexity of the phenomena 
discussed. Above all, let it be remembered that our object 
in building up a connected idea of the universe from the 
facts of observation is not to assert as unalterable truth 
the views we arrive at, but, by means of working hypotheses, 
to assist the mind to grasp the interrelations of the facts, 
and to prepare the way for a further advance. When we 
look back on the many transformations that theories in all 
departments of science have undergone in the past, we 



m =*: 



-Galactic 
Plane 



FIG. 1. Hypothetical Sect on of the Stellar System. 

shall not be so rash as to suppose that the mystery of the 
sidereal universe has yielded almost at the first attack. 
But as each revolution of thought has contained some 
kernel of surviving truth, so we may hope that our 
present representation of the universe contains something 
that will last, notwithstanding its faulty expression. 

It is believed that the great mass of the stars with 
which we are concerned in these researches are arranged 
in the form of a lens- or bun-shaped system. That is to 
say, the system is considerably flattened towards one plane. 
A general idea of the arrangement is given in Fig. 1 , where 
the middle patch represents the system to which we are 
now referring. In this aggregation the Sun occupies a fairly 
central position, indicated by -K The median plane of the 
lens is the same as the plane marked out in the sky by the 



32 STELLAR MOVEMENTS CHAP. 

Milky Way, so that, when we look in any direction along 
the galactic plane (as the plane of the Milky Way is called), 
we are looking towards the perimeter of the lens where the 
boundary is most remote. At right angles to this, that is, 
towards the north and south galactic poles, the boundary is 
nearest to us ; so near, indeed, that our telescopes can 
penetrate to its limits. The actual position of the Sun is 
a little north of the median plane ; there is little evidence 
as to its position with respect to the perimeter of the 
lens ; all that we can say is that it is not markedly 
eccentric. 

The thickness of the system, though enormous com- 
pared with ordinary units, is not immeasurably great. No 
definite distance can be specified, because it is unlikely 
that there is a sharp boundary ; there is only a gradual 
thinning out of the stars. The facts would perhaps be 
best expressed by saying that the surfaces of equal density 
resemble oblate spheroids. To give a general idea of the 
scale of the system, it may be stated that in directions 
towards the galactic poles the density continues practically 
uniform up to a distance of about 100 parsecs ; after that 
the falling off becomes noticeable, so that at 300 parsecs it 
is only a fraction (perhaps a fifth) of the density near the 
Sun. The extension in the galactic plane is at least three 
times greater. These figures are subject to large 
uncertainties. 

It seems that near the Sun the stars are scattered in a 
fairly uniform manner ; any irregularities are on a small 
scale, and may be overlooked in considering the general 
architecture of the stellar system. But in the remoter 
parts of the lens, or more probably right beyond it, there 
lies the great cluster or series of star-clouds which make 
up the Milky Way. In Fig. 1 this is indicated (in section) 
by the star-groups to the extreme left and right. These 
star-clouds form a belt stretching completely round the 
main flattened system, a series of irregular agglomera- 



ii GENERAL OUTLINE 33 

tions of stars of wonderful richness, diverse in form and 
grouping, but keeping close to the fundamental plane. 
It is important to distinguish clearly the two properties of 
the galactic plane, for they have sometimes been con- 
fused. First, it is the median plane of the bun-shaped 
arrangement of the nearer stars, and, secondly, it is the 
plane in which the star-clouds of the Milky Way are coiled. 

Not all the stars are equally condensed to the galactic 
plane. Generally speaking, the stars of early type con- 
gregate there strongly, whereas those of late types are 
distributed in a much less flattened, or even in a practically 
globular, form. The mean result is a decidedly oblate 
system ; but if, for example, we consider separately the 
stars of Type M, many of which are at a great distance 
from us, they appear to form a nearly spherical system. 

In the Milky Way are found some vast tracts of absorbing 
matter, which cut off the light of the stars behind. 
These are of the same nature as the extended irregular 
nebulae, which are also generally associated with the 
Milky Way. The dark absorbing patches and the faintly 
shining nebulae fade into one another insensibly, so 
that we may have a dark region with a faintly luminous 
edge. Whether the material is faintly luminous or not, 
it exercises the same effect in dimming or hiding the 
bodies behind it. There is probably some of this absorb- 
ing stuff even within the limits of the central aggregation. 
In addition to these specially opaque regions, it is probable 
that fine particles may be diffused generally through 
interstellar space, which would have the effect of dimming 
the light of the more distant stars ; but, so far as can be 
ascertained, this " fog " is not sufficient to produce any 
important effect, and we shall usually neglect it in the 
investigations which follow. 

In studying the movements of the stars we necessarily 
leave the remoter parts of space, confining attention mainly 
to the lens-shaped system, and perhaps only to the inner 

D 



34. STELLAR MOVEMENTS CHAP. 

parts of it, where the apparent angular movements are 
appreciable. Researches on radial motions need, not be 
quite so limited, because in them the quantity to be 
measured is independent of the distance of the stars ; 
but here too the nearer parts of the system obtain a 
preference, for observations are confined to the bright 
stars. Although thus restricted, our sphere of knowledge 
is yet wide enough to embrace some hundreds of thousands 
of stars (considered through representative samples) ; the 
results that are deduced will have a more than local 
importance. 

The remarkable result appears that within the inner 
system the stars move with a strong preference in two 
opposite directions in the galactic plane. There are 
two favoured directions of motion ; and the appearance 
is as though two large aggregates of stars of more or less 
independent origin were passing through one another, and 
so for the time being were intermingled. It is true that 
such a straightforward interpretation seems to be at 
variance with the plan of a single oblate system, which 
has just been sketched. Various alternatives will be 
considered later ; meanwhile it is sufficient to note that 
the difficulty exists. But, whatever may be the physical 
cause, there is no doubt that one line in the galactic 
plane is singled out, and the stars tend to move to and 
fro along it in preference to any transverse directions. 
We shall find it convenient to distinguish the two streams 
of stars, which move in opposite directions along the 
line, reserving judgment as to whether they are really 
two independent systems or whether there is some other 
origin for this curious phenomenon. The names assigned 
are 

Stream I. moving towards R. A. 94, Dec. +12'. 
II. ,, R.A. 274% Dec. -12. 

The relative motion of one stream with respect to 
the other is about 40 km. per sec. 



ii GENERAL OUTLINE 35 

The Sun itself has an individual motion with respect 
to the mean of all the stars. Its velocity is 20 kilo- 
metres per second directed towards the point R.A. 270 
Dec. + 35. The stellar movements that are directly 
observed are referred to the Sun as standard, and are 
consequently affected by its motion. This makes a 
considerable alteration in the apparent directions of the 
two streams ; thus we find 

Stream I. moving towards R.A. 91 C , Dec. - 15 ^ relatively to 
II. R.A. 288, Dec. -64 / the Sun ; 

and moreover the velocity of the first stream is about 1'8 
times that of the second (probably 34 and 19 km. per sec., 
respectively). Stream I is sometimes therefore referred to 
as the quick-moving stream, and Stream II as the slow- 
moving one ; but it must be remembered that this 
description refers only to the motion relative to the 
Sun. The stars which constitute the streams have, 
besides the stream-motion, individual motions of their 
own ; but the stream-motion sufficiently dominates over 
these random motions to cause a marked general agreement 
of direction. 

Stream I contains more stars than Stream II in the 
ratio 3 : 2. Though this ratio varies irregularly in different 
parts of the sky, the mixture is everywhere fairly 
complete. Moreover, there is no appreciable difference in 
the average distances of the stars of the two streams. It 
is not a case of a group of nearer stars moving in one 
direction across a background of stars moving the opposite 
way ; there is evidence that the two streams thoroughly 
permeate each other at all distances and in all parts of the 
heavens. 

A more minute investigation of this phenomenon shows 
that it is complicated by differences in the behaviour of 
stars according to their spectral type. An analysis which 
treats the heterogeneous mass of the stars as a whole 

D 2 



36 STELLAR MOVEMENTS CHAP. 

without any separation of the different types will fail to 
give a complete insight into the phenomenon. But, until 
a great deal more material is accumulated, this interrelation 
of stream motions and spectral type cannot be worked out 
very satisfactorily. The outstanding feature, however, is 
that the stars of the Orion Type (Type B) seem riot to 
share to any appreciable extent in the star-streaming 
tendency. Their individual motions, which are always 
very small, are nearly haphazard, though the apparent 
motions are, of course, affected by the solar motion. They 
thus form a third system, having the motion of neither of 
the two great streams, but nearly at rest relatively to the 
mean of the stars. This third system is not entirely 
confined to the B stars. In the ordinary analysis into 
two streams we always find some stars left over com- 
paratively few in number yet constituting a distinct 
irregularity which evidently belong to the same system. 
These stars may be of any of the spectral types. There is 
something arbitrary in this dissection into streams (which 
may be compared to a Fourier or spherical harmonic 
analysis of observations), and we can, if we like, adopt a 
dissection which gives much fuller recognition to this 
third system. 

At one time it seemed that the third stream, Stream 
as it is called, might be constituted of the very distai.t 
stars, lying beyond those whose motions are the main 
theme of discussion. If that were so, it would not be 
surprising to find that they followed a different law, and 
were not comprised in the two main streams. But this 
explanation is now found to be at variance with the facts. 
\\ have to recognise that Stream is to be found even 
among the nearer stars. 

The smallness of the individual movements of the 
B stars is found to be part of a much more general law. 
Astrophysicists have by a study of the spectra arranged 
the stars in what they believe to be the successive stages 



ii GENERAL OUTLINE 37 

of evolution. Now it is found that there is a regular 
progression in the size of the linear motions from the 
youngest to the oldest stars. It is as though a star was 
born without motion, and gradually acquires or grows one. 
The average individual motion (resolved in one direction) 
increases steadily from about 6*5 km. per sec. for Type B 
to 1 7 km. per sec. for Type M. 

We may well regard this relation of age and velocity as 
one of the most startling results of modern astronomy. 
For the last forty years astrophysicists have been studying 
the spectra and arranging the stars in order of evolution. 
However plausible may be their arguments one would have 
said that their hypotheses must be for ever outside the 
possibility of confirmation. Yet, if this result is right, 
we have a totally distinct criterion by which the stars are 
arranged in the same order. If it is really true that the 
mean motion of a class of stars measures its progress along 
the path of evolution, we have a new and powerful aid to 
the understanding of the steps of stellar development. 

It is not at all easy to explain why the stellar velocities 
increase with advancing development. I am inclined to 
think that the following hypothesis offers the best 
explanation of the facts. In a primitive state the star- 
forming material was scattered much as the stars are now, 
that is, densely along the galactic plane up to moderate 
distances, and more thinly away from the plane and at 
great distances. Where the material was rich, large stars, 
which evolved slowly, were formed ; where it was rare, 
small stars, which developed rapidly. The former are our 
early type stars ; not having fallen in from any great 
distance they move slowly and in the main parallel to the 
galactic plane. The latter our late type stars have 
been formed at a great distance, and have acquired large 
velocities in falling in ; moreover since they were not 
necessarily formed near the galactic plane, their motions 
are not so predominantly parallel to it. 



38 STELLAR MOVEMENTS CHAP. 

Whilst the individual motion of a star gradually increases 
from type to type, the stream- motion appears with remark- 
able suddenness. Throughout Type B, even up to B 8 and 
B 9, the two star-streams are unperceived ; but in the next 
type, A, the phenomenon is seen in its clearest and most 
pronounced form. Through the remaining types it is still 
very prominent, but there is an appreciable falling off. 
There is no reason to believe that this decline is due to 
any actual decrease in the stream-velocities ; it is only that 
the gradual increase of the haphazard motions renders the 
systematic motions less dominant. 

Among the most beautiful objects that the telescope 
reveals are the star-clusters, particularly the globular 
clusters, in which hundreds or even thousands of stars are 
crowded into a compact mass easily comprised within the 
field of a telescope. Recent research has revealed several 
systems, presumably of a similar nature to these, which 
are actually in our neighbourhood, and in one case even 
surrounding us. Being seen from a short distance the 
concentration is lost, and the cluster scarcely attracts notice. 
The detection of these systems relatively close to us is an 
important branch of study ; they are distinguished by the 
members having all precisely equal and parallel motions. 
The stars seem to be at quite ordinary stellar distances 
apart, and their mutual at traction is too weak to cause any 
appreciable orbital motion. They are not held together by 
any force ; and we can only infer that they continue to 
move together because no force has ever intervened to 
separate them. 

These " moving clusters" are contained within the central 
aggregation of stars. Many of the globular clusters, 
though much more distant, are probably also contained in 
it ; others, however, may be situated in the star-clouds of 
the Milky Way. Their distribution in the sky is curiously 
uneven ; they are nearly all contained in one hemisphere. 
They are most abundant in Sagittarius and Ophiuchus, 



ii GENERAL OUTLINE 39 

near a brilliant patch of the Milky Way, which is 
undoubtedly the most extraordinary region in the sky. 
This might be described as the home of the globular 
clusters. 

We shall also have to consider the nebulae, and their 
relation to the system of the stars. At this stage it 
may be sufficient to state that under the name "nebulae" 
are grouped together a number of objects of widely 
differing constitution ; we must not be deceived into 
supposing that the different species have anything in 
common. There is some reason for thinking that the spiral 
or "white" nebulas are objects actually outside the whole 
stellar system, that they are indeed stellar systems coequal 
with our own, and isolated from us by a vast intervening 
void. But the gaseous irregular nebulae, and probably 
also the planetary nebulae, are more closely associated with 
the stars and must be placed among them. 

It is now time to turn from this outline of the leading- 
phenomena to a more detailed consideration of the 
problems. The procedure will be to treat first the nearest 
stars, of which our knowledge is unusually full and direct. 
From these we pass to other groups that happen to be 
specially instructive. From this very limited number of 
stars, a certain amount of generalisation is permissible, 
but our next duty is to consider the motions of the stars 
in general ; this will occupy Chapters V. VII. After 
considering the dependence of the various phenomena on 
spectral type, we pass on to the problems of stellar 
distribution. This comes after our treatment of stellar 
motions, because the proper motions are, when carefully 
treated, among the most important sources of information 
as to the distances of stars. In Chapter XL we pass to 
subjects the Milky Way and Nebulae of which our 
knowledge is even more indefinite. The concluding Chapter 
attempts to introduce the problem of the dynamical forces 
under which motions of the stellar system are maintained. 



CHAPTER III 

THE NEAREST STARS 

MOST of our knowledge of the distribution of the stars 
is derived by indirect methods. Statistics of stellar magni- 
tudes and motions are analysed and inferences are drawn 
from them. In this Chapter, however, we shall consider 
what may be learnt from those stars the distances of which 
have been measured directly ; and, although but a small 
sample of the stellar system comes under review in this 
way, it forms an excellent starting point, from which 
we may proceed to investigations that are usually more 
hypothetical in their basis. 

For a parallax-determination of the highest order of 
accuracy, the probable error is usually about 0"'01. Thus 
the position of a star in space is subject to a compara- 
tively large uncertainty, unless its parallax amounts to at 
least a tenth of a second of arc. How small a ratio such 
stars bear to the whole number may be judged from the 
fact that the median parallax of the stars visible to the 
naked eye is only 0"'008 ; as many naked-eye stars have 
parallaxes below this figure as above it. We are therefore 
in this Chapter confined to the merest fringe of the 
surrounding universe, making for the present no attempt 
to penetrate into the general mass of the stars. 

Table 3 shows all the stars that have been found to have 

40 



CH. Ill 



THE NEAREST STARS 



parallaxes of 0"*20 or greater.* Only the most trustworthy 
determinations have been accepted, and in most cases at least 
two independent investigators have confirmed one another. 
The list is based mainly on Kapteyn and Weersma's com- 
pilation. 1 

TABLE 3. 

The Nineteen Nearest Stars. 
(Stars distant less than five parsecs from the Sun.) 



Star. 


Magni- 
tude. 


Spectrum. 


Parallax. 


Luminos- 
ity 

(Sun = l). 


Remarks. 


Groombridge 34 . 
TJ Cassiopeiae . . . 
T Ceti 


8-2 
3-6 
3-6 


Ma 

F8 
K 


0-28 
0-20 
0-33- 


o-oio 

1-4 
0-50 


Binary 
Binary 


f Eridani . ... 
CZ5 h 243 . . . 
Sirius 


3-3 
8-3 
- 1-6 


K 

G-K 

A 


0-31 
0-32 

0'38 


0-79 
0-007 
48-0 


Binary 


Procyon . ... 
Lai. 21185 . . . 
Lai. 21258 . . . 
OA (N.) 11677 . . 

a Centauri .... 

A (N.) 17415 . . 
Pos. Med. 2164 . . 
v Draconis .... 
a Aquilne .... 
61 Cygni 
f Indi 


0-5 
7-6 
8-9 
9-2 

0-3 

9-3 

8-8 
4-8 
0-9 
5-6 

4'7 


F5 
Ma 
Ma 

G, K5 

F 
K 

-rr 

A5 
K5 
K5 


0-32- 
0-40- 
0-20 
0-20 

0-76' 

0-27 
0-29 
0-20 
0-24 
0-31 
0-28 


97 
0-009 

0-011 

0-008 
J2-0 | 
10-6 J 
0-004 
0-006 
0-5 
12-3 

o-io 

0-25 


Binary 

Binary 
Binary 

Binary 


Kriiger 60 .... 
Lacaille 9352 . . . 


9-2 

7'4 


Ma 


0-26 
0-29 


0-005 
0-019 


Binary 



It is of much interest to inquire how far this list of 
nineteen stars is exhaustive. Does it include all the stars 
in a sphere about the Sun as centre with radius five 
parsecs ? In one respect the Table is admittedly incom- 
plete ; for stars fainter than magnitude 9'5 (on the B.D. 
scale), determinations are entirely lacking. A 9" U 5 star 
with a parallax 0"'2 would have a luminosity O'OOG, the 

* In using a table of this kind I am following the Astronomer Royal, 
F. \\ . Dyson, who first showed me its importance. 



42 STELLAR MOVEMENTS CHAP. 

Sun being the unit ; so that, in general, stars giving less 
than l/200th of the light of the Sun could not be 
included in the list. The distribution of the luminosities 
in column five of the Table leads us to expect that these 
very feeble stars may be rather numerous. 

Admitting, then, that Table 3 breaks off at about 
luminosity O'OOG, and that in all probability numerous 
fainter stars exist within the sphere, how far is it complete 
above this limit ? Generally speaking, stars are selected 
for parallax-determinations on account of their large 
proper motions. Most of the very bright stars have also 
been measured, but in no case has a parallax greater than 
0"'2 been found which was not already rendered probable 
by the existence of a large proper motion. To form some 
idea of the completeness with which the stars have been 
surveyed for parallax, consider those stars the motions of 
which exceed 1" per annum. F. W. Dyson has given a list 
of ninety-five of these stars,' 2 and it is probable that his list 
is nearly complete, at least as far as the ninth magnitude ; 
the Durchmusterungs and Meridian Catalogues of most 
parts of the sky have involved so thorough a scrutiny, 
that it would be difficult for motions as large as this to 
remain unnoticed. Of these ninety-five stars, sixty-five 
may be considered to have well-determined parallaxes, or 
at least the determinations have sufficed to show that they 
lie beyond the limits of our sphere ; among the former are 
seventeen of the nineteen stars of Table 3. For the 
remaining thirty, either measurements have been unat- 
tempted, or the determinations do not negative the possi- 
bility of their falling within the sphere. This remainder 
is not likely to be sa rich in large parallaxes, because it 
includes a rather large proportion of stars which only just 
exceed the annual motion of 1" ; but there are some 
notable exceptions. The star Cordoba 32416, mag. 8*5, 
having the enormous annual motion of G"'07 seems to 
have been left alone entirely. It may be expected that 



in THE NEAREST STARS 43 

further examination of these thirty stars will yield four or 
five additional members for our Table. 

Of stars with annual proper motions less than 1" the 
Table contains only two. It is not difficult to show that 
this is an inadequate proportion. The median parallax of 
stars distributed uniformly through the sphere (of radius 
5 parsecs) is 0"'25 ; now for a star of that parallax an 
annual motion of 1" would be equivalent to a linear 
transverse motion of 20 km. per sec. Approximately 
then for our sphere 

No. of PM's > 1" No. of transverse motions > 20 km. /sec. 



No. of PM's < 1" No. of transverse motions < 20 km. /sec. 

Now our general knowledge of stellar velocities, derived 
from other sources, is probably sufficiently good to give 
a rough idea of the latter ratio ; for it may be expected 
that the distribution of linear velocities within the sphere 
will not differ much from the general distribution outside. 
Taking the average radial velocity of a star as 17 km. 
per sec. (the figure given by Campbell for types K and M, 
which constitute the great majority of the stars), a 
Maxwellian distribution would give 64 per cent, of the 
transverse motions greater than 20 km. per sec. and 36 
per cent, less a ratio of 1*8 : 1. The addition of the 
solar motion will increase the proportion of high velocities ; 
and in those parts of the sky where it has its full effect 
the ratio is nearly 3:1. Probably we shall not be far 
wrong in assuming that there will be two-fifths as many 
stars with motions below 1" per annum as above it. 

Having regard to these considerations, the calculation 
stands thus 



No. of stars in the Table with proper motion greater than 1" . 17 

Proportionate allowance for stars not yet examined 5 

Due proportion with proper motion less than 1", say .... 9 

The Sun 1 

Total 32 



44 STELLAR MOVEMENTS CHAP. 

To this must be added an unknown but probably consider- 
able number of stars the luminosity of which is less than 
1 /200th of the Sun. 

In round numbers we shall take thirty as the density 
of the stars within the sphere (tacitly ignoring the 
intrinsically faint stars). As twenty of these are actually 
identified, the number may be considered to rest on 
observation with very little assistance from hypothetical 
considerations. 

This short list of the nearest stars well repays a careful 
study. Many of the leading facts of stellar distribution 
are contained in it; and, although it would be unsafe to 
generalise from so small a sample, results are suggested 
that may.be verified by more extensive studies. 

Perhaps the most striking feature is the number of 
double stars. It will be seen that eight out of the nine- 
teen are marked " binary." Why some stars have split 
into two components, whilst others have held together, is 
an interesting question ; but it appears that the fission of 
a star is by no means an abnormal fate. The stars which 
separate into two appear to be not much less numerous 
than those which remain intact. The large number of 
discoveries of variable radial velocity made with the 
spectroscope confirms this inference, though the spectro- 
scopists generally do not give quite so high a proportion. 
\V. W. Campbell 3 from an examination of 1600 stars 
concludes that one-quarter are spectroscopic binaries. But 
this proportion must be increased if visual binaries are 
included (for these are not usually revealed by the spectro- 
scope) ; and, in addition, there must be pairs too far 
separated to be detected as spectroscopic binaries, but too 
distant from us to be recognised visually. E. B. Frost, 
examining the stars of Type B, found that two-fifths of 
those on his programme were binary ; he also found that 
in Boss's Taurus cluster the proportion was one-half. 4 



in THE NEAREST STARS 45 

In the Ursa Major cluster nine stars out of fifteen are 
known to be binary.' Apparently the division into two 
bodies takes place at a very early stage in a star's 
history or in the pre-stellar state, as is evidenced by the 
high proportion found in the earliest spectral type. As 
time passes the components separate further from each 
other and the orbital velocity becomes small, so that in the 
later types an increasing proportion escapes detection. 
There thus seems no reason to doubt that the proportion 
eight out of nineteen very fairly represents the general 
average, but at the very lowest it cannot be less than one 
in three. 

The luminosities of the stars in the Table range from 
48 to 0*004, that of the Sun being taken as unit. We 
have already seen that the lower limit is due to the fact 
that our information breaks off near this point, and it is 
natural to expect that there must be a continuous series of 
fainter bodies terminating with totally extinct stars. At 
the other end of the scale a larger sample would un- 
doubtedly contain stars of much greater luminosity ; but 
these are comparatively rare in space. Arcturus, for 
example, is from 150 to 350 times as bright as the Sun, 
An tares at least 180 times, whilst Eigel and Canopus can 
scarcely be less than 2000 times as bright, allowing in 
each case a wide margin for the possible uncertainty of the 
measured parallaxes. There is little doubt that these 
estimates err on the side of excessive caution. 

For a star of the same intrinsic luminosity as the Sun 
to appear as bright as the sixth magnitude, its parallax 
must be not less than 0"'08. Since there is no doubt that 
the majority of stars visible to the naked eye are much 
further away than this, it follows that the great majority 
of these stars must be much brighter, in fact more than a 
hundred times as bright as the Sun. We might hastily 
suppose that the Sun is therefore far below the average 
brilliancy. But Table 3 reveals a very different state of 



4 6 STELLAR MOVEMENTS CHAP. 

things. Of the nineteen stars, only five exceed it, whilst 
fourteen are fainter. The apparent paradox directs attention 
to a fact which we shall have occasion to notice frequently. 
The stars visible to the naked eye, and the stars 
enumerated in the catalogues, are quite unrepresentative 
of the stars as a whole. The more intensely luminous 
stars are seen and recorded in numbers out of all propor- 
tion to their actual abundance in space. It is of great 
importance to bear in mind this limitation of statistical 
work on star- catalogues ; we ought to consider whether 
the results derived from the very special kind of stars that 
appear in them may legitimately be extended to the stars 
as a whole. 

In the same way, the Table gives a very different idea 
of the proportions in which the different types of spectra 
occur from the impression we should gather by examining 
the catalogues. Four stars of Type M are included, 
although in the catalogues this class forms only about a 
fifteenth of the whole number. On the other hand, Type 
B stars (the Orion type), which are rather more numerous 
than Type M in the catalogues, have not a single repre- 
sentative here. The explanation is that these M stars are 
usually very feebly luminous objects (as may be seen in 
the Table), and can rarely be seen except in our immediate 
neighbourhood. Type B stars on the other hand are 
intensely bright, and although they occur but sparsely in 
space, we can record even those near the limits of the 
stellar system, making a disproportionately large number. 
Again in the catalogues the Sirian Type stars (A) and the 
Solar Type (F, G, K) are about equally numerous, but in 
this definite volume of space the latter outnumber the 
former by ten to two. 

In order to obtain more extensive data as to the true 
proportions of the spectral types, and the relation of 
spectral type to luminosity, Table 4 has been drawn up, 
containing the stars with fairly well-determined parallaxes 



Ill 



THE NEAREST STARS 



47 



between 0"'19 and 0"T1. These are not generally so trust- 
worthy as the parallaxes of Table 3, because chief atten- 
tion has naturally been lavished on those stars known to 
be nearest to us ; but the standard is fairly high, a great 
many measured parallaxes being rejected as too uncertain. 
Those marked with an asterisk are the best determined 
and may be considered equal in accuracy to the parallaxes 
of Table 3 ; but as the parallaxes are smaller, the propor- 
tionate uncertainty of the calculated luminosity is greater. 

TABLE 4. 
Stars distant between 5 and 10 parsecs from the Sun. 



Star. ^ludT 


Spectrum. 


Annual 
Proper 
Motion. 


Parallax. 


Luminos- 
ity 
(Sun = l). 


f Toucanse . . . 4 "3 


F8 


2-07 


0-15 


1-3 


*,* Hydri . ... 2'9 


G 


2-24 


0-14 


5-4 


54 Piscium ... O'l 


K 


0-59 


0-15 


0-26 


Mayer 20 ... 5 '8 


K 


1-34 


0-16 


0-28 


*p. Cassiopeire ... 5 '3 


G5 


375 


0-11 


1-0 


8 Trianguli ... 5'1 


G 


1-16 


0-12 


1-0 


Pi. 2* 123 ... 5-9 


G5 


2-31 


0-14 


0-33 


e Eridani ... 4 '3 


G5 


3-15 


0-16 


1-15 


*5 Eridani ... 3'3 


K 


075 


0-19 


2-1 


o 2 Eridani .... 4 '5 


G5 


4-08 


0-17 


0-84 


X Aurig&e .... 4'8 


G 


0-85 


0-11 


1-5 


*Weisse 5 h 592 . . 8 '9 


Ma 


2-23 


0-18 


0-013 


Pi. 5 h 146 .... 6-4 


G2 


0-55 


0-11 


0-32 


*Fed. 1457-8 ... 7 "9 


Ma 


1*69 


0-16 


0-042 


*Groombridge 1618. 6 '8 


K 


1-45 


0-18 


0-09 


43 Coma* .... 4'3 


G 


1-18 


0-12 


2-2 


*Lalande 25372 . . 87 


K 


2-33 


0-18 


0-017 


Lalande 26196 . . 7'6 


G5 


0-68 


0-14 


0-074 


*Pi. 14 h 212 ... 5-8 


K 


2-07 


0-17 


0-26 


*Gronin^en VII., 










No. 20 .... 107 





1-22 


0-13 


0-005 


fHerculis .... 3'0 


G 


0-61 


0-14 


5-0 


*Weisse 17 h 322 . . 7 '8 


Ma 


1-36 


0-12 


0-08 


70 Ophiuchi ... 4'3 


K 


1-15 


0-17 


1-1 


*17 Lyrae C. ... 11 '3 





175 


0-13 


0-003 


Fomalhaut .... 1*3 


A3 


0-37 


0-14 


25-0 


*Bradley 3077 - . . 5-6 


K 


2-11 


0-14 


0-45 


*Lalande 46650 . . 8'9 


Ma 


1-40 


0-18 


0-013 



Table 4 is far from being a complete list of the stars 
within the limits. There should be about 200 stars in 



48 STELLAR MOVEMENTS CHAP. 

this volume of space, but only 27 are given here. 
However, the incompleteness will not much affect the 
present inquiry, except that a number of the faintest 
stars, which are especially of Types K and Ma, will 
naturally be lost. This is noticeable when the list is 
compared with Table 3. 

Collecting the stars of different spectral types, we have 
the following distribution of luminosities from Tables 3 
and 4 : 

Luminosities. 



Type of Parallaxes Parallaxes 

Spectrum. greater than 0'20". from 0'19" to O'll". 

A ... 48-0 

A3 . . 25-0 

Ao . . 12-3 

F . . . 0-004 

F5 . . 9-7 

F8 . . 1-4 1-3 

G . . . 2-0 *5-4, 5-0, 2-2, 1-5, I'O 

G2 . . 0-32 

G5 . . 1-15, *1-0, 0-84, 0-33, 0'074 

K /0-79 0-5 0-5 0-OOfi I * 2 ' 1 r1 ' *' 45 ' ' 28 ' *' 26 

. . O < J, 5, o, . *. *. 



K5 . . 0-6, 0-25 

Ma . . 0-019, 0-011 0-010, 0*009 *0'08, *0'042, *0'013, *0'013 

This summary shows a remarkable tendency towards 
equality of brightness among stars of the same type, and 
there is a striking progressive diminution of brightness 
with advance in the stage of evolution. The single star 
of Type F (strictly FO) makes a curious exception. This 
star, OA (N) 17415, was measured with the heliometer 
by Kriiger as early as 1863 ; apparently the determination 
was an excellent one. It would perhaps be desirable to 
check his result by observations according to more modern 
methods ; but we are inclined to believe that the exception 
is real. 

It would be tempting to conclude that the great range 
in absolute luminosity of the stars is mainly due to 
differences of type, and thnt within the same spectral 

parallaxes. 



in THE NEAREST STARS 49 

the range is very limited. We might apply this in 
searching for large parallaxes ; for it would seem that, 
since stars of Type Ma have so far been found to be of 
very feeble luminosity, any star of that class which 
appears bright must be very close to us. This hope is 
not fulfilled. The bright Ma stars which have been 
measured Betelgeuse, Antares, 77 Geminorum and 8 Vir- 
ginis have all very small parallaxes, and are certainly much 
more luminous even than Sirius, the brightest star in the 
Tables. It is perhaps unfortunate that these brilliant 
exceptions force themselves on our notice, whilst the far 
greater number of normal members of the class are too 
faint to attract attention ; we must not be led into an 
exaggerated idea of the number of these luminous third 
type stars. But it is clear that, notwithstanding the 
tendency to equality, if a large enough sample is taken, 
the range of luminosity is very great indeed. We shall 
have to return to this subject in Chapter VIII. 

Table 5 contains some details as to the motions of the 
nineteen nearest stars. The transverse velocities are formed 
by using the measured parallaxes to convert proper motions 
into linear measure.* The radial motions from spectroscopic 
observations are added when available. These motions are 
relative to the Sun ; if we wished to refer them to the 
centroid of the stars, we should have to apply the solar 
motion of 20 km. per sec., which might either increase or 
decrease the velocities according to circumstances, but 
would usually somewhat decrease them. After allowing 
for this, the commonness of large velocities is still a strik- 
ing and most surprising feature. The ordinary studies of 
stellar motions do not lead us to expect anything of the 
kind ; and in fact it is not easy to reconcile the general 

* The formula, which is often useful, is 

Linear speed = annual pjgpermotion_ x 4 . ;4 km per sec 
parallax 

E 



STELLAR MOVEMENTS 



CHAP. 



investigations with the results of this special study of a 
small collection of stars. Taking the fastest moving stars, 
Type M, Campbell found an average radial motion of 17 
km. per sec. Assuming a Maxwellian distribution of 
velocities, this would give for a transverse motion (i.e., 
motion in two dimensions) 



Speed greater than 



60 km. per sec. 
80 ,, 

100 ,, 



1 star in 53 
1 star in 1,100 
1 star in 60,000 



The presence of 3 stars in the list with transverse 
velocities of more than 1 00 km. per sec. (and therefore 
certainly more than 80 km. per sec., when the solar motion 
is removed) is wholly at variance with the above statistical 
scheme. 

TABLE 5. 
Motions of the nineteen nearest stars. 



Star 


Proper 5 


lotion. 


Radial 






Arc. 


Linear. 


Velocity. 




Groombridge 34 .... 
r\ Cassiopeiae 


2-85 
1-25 


km. per sec. 
48 
30 


km. per sec. 
+ 10 


I. 
I. 


T Ceti . 


1-93 


28 


-16 


II. 


f Eridani 


1-00 


15 


+ 16 


II. 


CZ 5 h 243 .. 


870 


129 


+ 242 


II. 


Sirius 


1-32 


16 


7 


II 


Prooyon 
Lalande 21185 
Lalande 21258 
OA(N) 11677 
a Centauri 


1-25 
477 
4-46 
3-03 
3-66 


19 
57 
106 
72 
23 


-3 
-22 


1.1 

II. 
I. 
I. 
I. 


OA (N) 17415 .... 
Pos. Med. 2164 . . . 
<r Draconis 


1-31 

2-28 
1-84 


23 
37 
43 


+ 25 


II. 
I. 
II. 


a Aquilae 


0-65 


T3 


-33 


I. 


61 Cygni 
( Indi . . . . . 


5-25 
4'67 


80 
79 


-62 
-39 


I. 
I 


Kriiger 60 
Lacaille 9352 


0-92 
7-02 


17 
115 


+ 12 


II. 
I. 













We cannot attribute the result to errors in the paral- 
laxes, for if it should happen that the parallax has 



in THE NEAREST STARS 51 

been overestimated, the speed will have been under- 
estimated ; and it is scarcely likely that any of these 
stars have parallaxes appreciably greater than those 
assigned in the Table. A possible criticism is that these 
stars have been specially selected for parallax-measurement, 
because they were known to have large proper motions ; 
but the objection has not much weight unless it is seriously 
suggested that there are, in this small volume, the hundreds 
or even thousands of stars of small linear motion that the 
statistical scheme seems to require. Moreover we have 
already shown that on the ordinary view as to stellar 
motions, seven additional stars would supply the loss due 
to the neglect of stars with motions less than 1" per 
annum. 

Nor can we help matters by throwing over the Max- 
wellian law. Originally used as a pure assumption, this 
law has been confirmed in the main by recent study of the 
radial motions. We should, however, be prepared to admit 
that it may not give quite sufficient very large motions. 
It has long been known that certain stars, as Arcturus and 
Groombridge 1830, had excessive speeds that seemed to 
stand outside the ordinary laws. But we find that the 
average transverse motion of the nineteen stars is fifty km. 
per sec. ; this considerably exceeds the average speed we 

should deduce from the stars that come into the ordinary 

/ 

investigations. In round numbers, a mean speed of thirty 
km. per sec. relative to the Sun would have been 
expected.* 

Thus once more it is found that the survey of stars 
in the limited volume of space very near to the Sun leads 
to results differing from those derived from the stars of 
the catalogues. We may fall back on the same explana- 

* An average radial speed of 17 km. per sec. gives an average transverse 
speed of 26 '5 km. per sec., the factor being ^ whatever the law (Maxwel- 

lian or otherwise) of stellar motions. This does not include the solar motion, 
which, however, would not increase the result very greatly. 

E 2 



52 STELLAR MOVEMENTS CHAP. 

tion as before, that the catalogues give a very untypical 
selection of the stars. But this time the result is more 
surprising ; it would scarcely have been expected that 
the catalogue-selection, which is purely by brightness, 
would have so large an effect on the motions. Yet this 
seems to be the case. We notice that the three stars 
with transverse speeds of more than 100 km. per sec. have 
luminosities 0'007, O'Oll and 0'019. These would be 
far too faint to come into the ordinary statistical 
investigations. The five stars brighter than the Sun have 
all very moderate speeds. Setting the nine most lum- 
inous stars against the ten feeblest, we have 

Luminosity. Mean transverse speed. 

9 Brightest stars . . 48 '0 to 0'25 29 km. per sec. 

10 Faintest . . O'lO 0'004 68 

It is the stars with luminosity less than l/10th that 
of the Sun, with which we are scarcely ever concerned 
in the ordinary researches on stellar motions, that are 
wholly responsible for the anomaly. The nine bright 
stars simply confirm our general estimate of thirty km. 
per sec. for the average speed. 

The stars of Table 4 also add a little evidence pointing 
in the same direction. The parallaxes are scarcely accurate 
enough (proportionately to their size) to be used for this 
purpose ; but we give the result for what it is worth. It 
may be noted that the parallaxes are likely to be a little 
overestimated and therefore both luminosities and speeds 
will be underestimated. On the other hand, the influence 
of selection (on account of large proper motion) will be 
greater than in Table 3, tending to increase the mean 
speed unduly. There are nine stars given in Table 4 as 
having luminosities less than O'l ; their mean speed is 
forty-eight km. per sec. Thus these stars have speeds con- 
siderably in excess of the thirty km. per sec. originally 
expected. They do not, however, include any excessive 
speeds. 



in THE NEAREST STARS 53 

It would be very desirable to have more evidence on 
this point before drawing a general conclusion ; but our 
task is to sum up the present state of our knowledge, 
however fragmentary. The stars, of which the proper 
motions and radial velocities are ordinarily discussed, are 
almost exclusively those at least as bright as the Sun. It is 
generally tacitly assumed that the motions of the far 
more numerous stars of less brilliance will be similar to 
them. But the present discussion affords a strong sus- 
picion that there exists a class of stars, comprising the 
majority of those having a luminosity below OT, the speeds 
of which are on the average twice as great as the fastest 
class ordinarily considered. Apparently the progressive 
increase of velocity with spectral type does not end with 
the seventeen km. per sec. of the brilliant members of 
Type M, but continues for fainter stars up to at least twice 
that speed. 

In the final column of Table 5. each star has been 
assigned to its respective star-stream according to the 
direction in which it is moving. We see that eleven stars 
probably belong to Stream I, and eight probably to 
Stream II. This is in excellent accordance with the ratio 
3 : 2 derived from the discussion of the 6000 stars of 
Boss's Catalogue. We are not able to detect any significant 
difference between the luminosities, spectra, or speeds of 
the stars constituting the two streams. The thorough 
inter-penetration of the t\vo star-streams is well illustrated, 
since we find even in this small volume of space that 
members of both streams are mingled together in just 
about the average proportion. 

REFERENCES. CHAPTER III. 

1. Kapteyn and Weersma, Groningen Publications, No. 24. 

2. Dyson, Proc. Roy. Sac. Edtnluruh, Vol. 29, p. 378. 
:5. Campbell, Stellar Motions, p. 245. 

4. Frost, Axh-nit/iit.-ii'rtd J<i,-,i<i.l, Vol. 29, p. 237. 

5. Hertzsprung, Attrophyrical Journal, Vol. 30, p. 139. 



CHAPTER IV 

MOVING CLUSTERS 

THE investigation of stellar motions has revealed a 
number of groups of stars in which the individual 
members have equal and parallel velocities. The stars 
which form these associations are not exceptionally near to 
one another, and indeed it often happens that other stars, 
not belonging to the group, are actually interspersed 
between them. We may perhaps arrive at a better 
understanding of these systems by recalling a few 
elementary considerations regarding double stars. 

In only a small proportion of the double stars classed 
as " physically connected " pairs, has the orbital motion of 
one component round the other been detected. In most 
cases the connection is inferred from the fact that the 
two stars are moving across the sky with the same proper 
motion in the same direction. The argument is that, 
apart from exceptional coincidences, the equality of angular 
motion signifies both an equality of distance and an 
equality of linear velocity. Accordingly the two stars 
must be close together in space, and their motions are such 
that they must have remained close together for a long 
period. Having established the fact that they are 
permanent neighbours, we may rightly deduce that their 
mutual gravitation will involve some orbital motion, though 
it may be too slow to detect ; but that is a subsidiary 



CH. iv MOVING CLUSTERS 55 

matter, and, in speaking of physical connection, we are not 
thinking of two stars tied together by an attractive force. 
The connection, if we try to interpret it, appears to be 
one of origin. The components have originated in the same 
part of space, probably from a single star or nebula ; they 
started with the same motion, and have shared all the 
accidents of the journey together. If the path of one is 
being slowly deflected by the resultant pull of the stellar 
system, the path of the other is being deflected at the 
same rate, so that equality of motion is preserved. It is 
true that the mutual attraction in these widely separated 
binaries may help to prevent the stars separating ; but it 
is a very feeble tie, and, in the main, community of motion 
persists because there are no forces tending to destroy 
it. 

From this point of view we may have physically con- 
nected pairs separated by much greater and even by 
ordinary stellar distances, remembering, however, that the 
greater the distance the more likely are they to lose their 
common velocity by being exposed to different forces. It 
is known that exceedingly wide pairs do exist The case 
of A Ophiuchi and Bradley 2179 may be instanced ; these 
stars are separated by about 14', but have the same 
unusually large motion of l r/< 24 per annum in the same 
direction. In general it would be difficult to detect pairs 
of this kind ; for unless the motion is in some way 
remarkable, an accidental equality of motion must often 
be expected, and it would be impossible to distinguish the 
true pairs from the spurious. It is only when there is 
something unusual in the amount or in the direction of 
the motion that there are grounds for believing the equality 
is not accidental. 

In the Moving Clusters we find a closely similar kind of 
physical connection. They are considerable groups of 
stars, widely separated in the sky, but betraying their 
association by the equality of their motions. The most 



56 STELLAR MOVEMENTS CHAP. 

thoroughly investigated example of a moving cluster is 
the Taurus-stream, which comprises part of the stars of 
the Hyades and other neighbouring stars. The existence 
of a great number of stars with associated motions in this 
region was pointed out by R. A. Proctor ; but the researches 
of L. Boss have shown the nature of the connection in 
a new light. Thirty-nine stars are recognised as belong- 
ing to the group, distributed over an area of the sky about 
15 square; there can be no doubt that many additional 
fainter stars l in the region also belong to the cluster but, 
until better determinations of their motions have been 
made, these cannot be picked out with certainty. 

The first criterion in such a case as this is that the motions 
should all appear to converge towards a single point in the 
sky ; in Fig. 2 the arrows indicate the observed motions 
of these stars, and the convergence is well shown. From 
this it may be deduced that the motions are parallel ; for 
lines parallel in space appear, when projected on a sphere, 
to converge to a point. It is true that the same appear- 
ance would be produced if the motions were all converging 
to or diverging from a point, but either supposition is 
obviously improbable. Theoretically there may be a slight 
degree of divergence, the cluster having been originally 
more compact ; but calculation shows that on any 
reasonable assumption as to the age of the cluster the 
divergence must be quite negligible. As the stars are not 
all at the same distance from the Sun, the fact that the 
speeds are all equal cannot be demonstrated in the same 
exact way ; but, allowing for the foreshortening of the 
apparent motion in the front of the cluster as compared 
with the rear, the proper motions all agree with one 
another very nearly. The divergences are just what 
we should expect if the cluster extends towards the Sun 
and away from it to the same distance that it extends 
laterally. 

It is clear that in a cluster of this kind the equality ;md 



IV 



MOVING CLUSTERS 



57 





c 


J C 


si C 
sJ C 


} c 
a * 


i 


3 T 


-c 


J -c 


3 


"u 


3 








































\ 














"T 
CJ 












\ 














ID 
^ 










\ 














I 










\ 






\ 

\\ 


I 








f 


ID 






V 


s 


\ 

\ 


5 


-\M 


\ 
. \ 










T 














*! 

* 


\ V 










CM 
CM 














i 


. 
1 

\ 


| 




I 




CO 












\\ 








\ 








T 

oc 










' 


\ 


\ \ 




\ 








TT 
U3 












\ 


\ 




\ 


I 


i 
1 




in 
> 














\ 


\ 


1 








CV 














\ 


t 


! 

























\ 


\ 


1 

\ 






C'J 

oc 
















\ 




\ 


i 




CJ 

ur 


















\ 


1 
1 

4 4 \ 
\ \ 


1 




CO 

-r 




















V x \ 






rr 
CM 




















\ \ 


; 




in 






















1 




> 























































58 STELLAR MOVEMENTS CHAP. 

parallelism of the motions must be extremely accurate, 
otherwise the cluster could not have held together. 
Suppose that the motion of one member deviated from the 
mean by one km. per sec. ; it would draw away from the 
rest of the cluster at the rate of one astronomical unit in 
4f years. In ten million years it would have receded ten 
parsecs (the distance corresponding to a parallax of O^'IO). 
We shall see later that the actual dimensions of the cluster 
are not so great as this ; the remotest member is about 
seven parsecs from the centre. According to present 
ideas, ten million years is a short period in the life even of 
a planet like the earth ; the age of the Taurus-cluster, 
which contains stars of a fairly advanced type of evolution, 
must be vastly greater than this. From the fact that it 
still remains a compact group we deduce that the 
individual velocities must all agree to within a small 
fraction of a kilometre per second. 

The very close convergence of the directions of motion 
of these stars supports this view ; the deviations may all 
be attributed to the accidental errors of observation. In 
fact the mean deviation (calculated - observed) in position 
angle is 1'8, whereas the expected deviation, due to the 
probable errors of the observed proper motions, is greater 
than this, a paradox which is explained by the fact that 
stars for which the accidental error is especially great 
would not be picked out as belonging to the group. 

To complete our knowledge of this cluster, one other 
fact of observation is required ; namely, the motion in the 
line of sight of any one of the individual stars. Actually 
six have been measured, and the results are in satisfac- 
tory accordance. The data are now sufficient to locate 
completely not only the cluster but its individual 
members and also to determine the linear motion, 
which, as has been shown, must be the same for all the 
stars to a very close approximation. This can be done as 
follows : 



IV 



MOVING CLUSTERS 



59 




The position of the -convergent point, shown at the 
extreme left of Fig. 2, is found to be 

R.A. 6 h 7 m '2 Dec. +6' 56' (1875'0) 

with a probable error of 1*5 chiefly in right ascension. 
If is the observer (Fig. 3), let 
OA be the direction of this con- 
vergent point. 

Consider one of the stars S of 
the cluster. Its motion * in space 
ST must be parallel to OA. Re- 
solve ST into transverse and radial 
components SX and SY. If SY 
has been measured by the spec- 
troscope, we can at once find ST 
for 

ST = SY sec TSY O 

= SY sec ACS FIG. 3. 

and, since A and S are known points on the celestial 
sphere, the angle AOS is known. 

Since the velocity ST is the same for every star of the 
cluster, it is sufficient to determine it from any one 
member the radial velocity of which has been measured. 
The result is found to be 45*6 km. per sec. 

We next find the transverse velocity for each star, 
which is equal to 

45 '6 sin AOS km. per sec. 

And the stars' distances are given by 

transverse velocity = distance x observed proper motion 

when these are expressed in consistent units. 

It will be seen that the distance of every star is found by 
this method, not merely those of which the radial velocity 
is known. The distance is found with a percentage accuracy 



* The motions considered here are all measured relatively to the Sun. 



60 STELLAR MOVEMENTS CHAP. 

about equal to that of the observed proper motion ; for the 
other quantities which enter into the formulae are very 
well-determined. As the proper motions of these stars 
are large and of fair accuracy, the resulting distances are 
among the most exactly known of any in the heavens. 
The parallaxes range from 0"*021 to 0"*031, with a mean 
of 0"*025. A direct determination by photography, the 
result of the cooperation of A. S. Donner, F. Kiistner, 
J. C. Kapteyn, and W. de Sitter, 2 has yielded the mean 
value 0"'023::0"'0025, which, though presumably less 
accurate than the result obtained indirectly, is a 
satisfactory confirmation of the legitimacy of Boss's- 
argument. 

From these researches the Taurus-cluster appears to be 
a globular cluster with a slight central condensation ; its. 
whole diameter is rather more than ten parsecs. The 
question arises whether this system can be regarded as- 
similar to the recognised globular clusters revealed by the 
telescope. If there were no more members than the thirty- 
nine at present known, the closeness of arrangement of the 
stars would not be greater than that which we have found in 
the immediate neighbourhood of the Sun. But it appears- 
that the 39 are all much brighter bodies than the Sun, and 
it is not fair to make a comparison with the feebly 
luminous stars discussed in the last chapter. According 
to a rough calculation the members of the Taurus-cluster 
may be classified as follows : 

5 stars with luminosity 5 to 10 times that of the Sun 

18 10 20 

11 20 50 

5 50 100 

In the vicinity of the Sun we have nothing to compare 
with this collection of magnificent orbs. These stars, it 
is true, are separated by distances of the usual order of 
magnitude ; but their exceptional brilliancy marks out 
this portion of space from an ordinary region. Whether 



iv MOVING CLUSTERS 61 

there are or are not other fainter members accompanying 
them, the term cluster is appropriate enough. There can 
be 110 doubt that, viewed from a sufficient distance, this 
assemblage would have the general appearance of a 
globular star-cluster. 

The known motion of the Taurus-cluster permits us to 
trace its past and future history. It was in perihelion 
800,000 years ago ; the distance was then about half what 
it is now. Boss has computed that in 65,000,000 years 
it will (if the motion is undisturbed) appear as an ordin- 
ary globular cluster 20' in diameter, consisting largely of 
stars from the ninth to the twelfth magnitude. 

It is interesting to note that a cluster of the size of this 
Taurus group must contain many interloping stars not 
belonging to it. Even if we omit the outlying members, 
the system fills a space equal to a sphere of at least 5 
parsecs radius. Now such a sphere in the neighbourhood 
of the Sun contains about 30 stars. We cannot suppose 
that a vacant lane among the stars has been specially left 
for the passage of the cluster. Presumably then the stars 
that would ordinarily occupy that space are actually there 
non-cluster stars interspersed among the actual members 
of the moving cluster. It is a significant fact that the 
penetration of the cluster by unassociated stars has not 
disturbed the parallelism of the motions or dispersed the 
members. 

The Ursa Major system is another moving cluster of 
which detailed knowledge has been ascertained. It has 
long been known that five stars of the Plough, viz., @, 7, 
S, e and { Ursae Majoris, form a connected system. By the 
work of Ejnar Hertzsprung it has been shown that a 
number of other stars, scattered over a great part of the 
sky, belong to the same association. The most interest- 
ing of these scattered members is Sirius ; and for it the 
evidence of the association is very strong. Its parallax 
and radial velocity are both well -determined, and agree 



62 STELLAR MOVEMENTS CHAP. 

with the values calculated from the motion of the whole 
cluster. The method by which the common velocity is 
found, and the individual stars are located in space, is the 
same as that employed for the Taurus-cluster. The 
velocity is 18'4 km. per sec. towards the convergent point 
R.A. 127'8, Dec. 4- 4 0* 2, when measured relatively to the 
Sun. When the solar motion is allowed for, the " absolute " 
motion is 28*8 km. per sec. towards R.A. 285, Dec. -2. 
As this point is only 5 from the galactic plane, the motion 
is approximately parallel to the galaxy. 

In Table 6 particulars of the individual stars are set 
down, including the parallaxes and radial velocities deduced 
by Hertzsprung from the known motion of the system. 
In most cases the calculated radial velocities have been 
confirmed by observation ; 3 but for nearly all the 
parallaxes, it has not been possible as yet to test the 
values given. It is quite likely that one or more stars 
have been wrongly included ; but there can be little doubt 
that the majority are genuine members of the group. The 
rectangular co-ordinates are given in the usual unit (the 
parsec), the Sun being at the origin, Oz directed towards 
the convergent point R.A. 127'8, Dec. +40'2, and Ox 
towards R.A. 307'8, Dec. + 49'8, so that the plane zOx 
contains the Pole. If a model of the system is made from 
these data, it is found, as H. H. Turner 4 has shown, that 
the cluster is in the form of a disk ; its plane being nearly 
perpendicular to the galactic plane. The flatness is very 
remarkable, the average deviation of the individual stars 
above or below the plane being 2'0 parsecs, a distance 
small in comparison with the lateral extent of the cluster, 
viz., 30*to 50 parsecs. In the last column are given the 
absolute luminosities in terms of the Sun as unit ; it is 
interesting to note that the three stars of Type F are the 
faintest, with luminosities 10, !>. ;md 7 respectively. 



IV 



MOVING CLUSTERS 



TABLE C>. 
The Ursa Major System. 



1 












Computed. 




Star. 




Spec-- 
trum. 


"i. v 




Rectangular 
Co-ordinates. 


Lumin- 
osity 
(8un=l). 


Paral- 


Radial 








lax. 


Velocity: 












// 


km./Bec. 


x y z 




3 Eridani . . . 2'92 


A2 0-034 


-7-5 


-13-8 -23-3 12-1 


96 


.d Auriga- . . . 2 '07 


Ap 0-024 


-iti-u 


7-8 -18-8 36-3 


410 


Shins .... 1*58 


A 0-387 


-8'5 


-2-0 -I'l 1-2 


46 


37 Tis;.- Maj. . 5-16 


F 0-045 


-16-6 


7'6 5-8 19-9 


7 


Urs Maj. . 2 '44 


A 


0-047 


-16-1 7'6 6-9 18-7 


76 


8 Leonis ... 2 '58 


A 2 


0-084 


-14-4 -2-3 7-1 9-3 


21 


y Urste Ma]. . 2'.~4 


A 0-042 


-15-0 


8-9 10-5 19-3 


87 


8 Urste Maj. . . 3 '44 


A2 


0-045 


-14-4 


9-8 97 17-2 


32 


Groom. 1930 . 


5-87 


F 0-028 


-13-4 


18-6 15-1 257 


9 


Urs Mai. - 1'68 


Ap 


0-042 


-13-2 


11-4 117 16-9 


190 


78 Ursjt Maj. . 4 '89 


F 0-042 


-13-0 


12-0 11-8 16-8 


10 


rrsa? Maj. . . 


/2-40 
\3-96 


~v?| ' 043 


-12-2 


11-9 12-5 15-3 


J93 
) 22 


a Coronse . . . 


2-31 


A 


0-041 


-2-2 


12-0 20-9 2-9 


110 



The stars of the Orion type of spectrum present several 
examples of moving clusters. In the Pleiades we have an 
evident cluster, in the ordinary sense of the term, and, as 
might be expected, the motions of the principal stars and 
at least fifty fainter stars are equal and parallel.^ The bright 
stars of the constellation Orion itself (with the exception 
of Betelgeuse, the spectrum of which is not of TypeB) also 
appear to form a system of this sort, the evidence in this 
case being mainly derived from their radial velocities, since 
the transverse motions are all exceedingly small. In 
Orion a faint nebulosity forming an extension of the 
Great Orion Nebula, has been discovered, which appears to 
fill the whole region occupied by the stars ; it probably 
consists of the lighter gases and other materials not yet 
absorbed by the stars which are developing. The velocity 
of the nebula in the line of sight agrees with that of the 

* This is the case as regards the proper motions. The radial velocities of 
the six brightest stars show some surprising differences (Adams, Astro- 
phijsical Journal, Vol. 19, p. 338), but owing to the difficult nature of the 
spectra the determinations are not very trustworthy. 



STELLAR MOVEMENTS 



CHAP. 



stars of the constellation. A similar nebulosity is found in 
the Pleiades. 

In the case of these, the youngest of the stars, the 
argument by which we deduced the accurate equality of 
motion in the Taurus-cluster scarcely applies; particularly 
in Orion, the dimensions of which must be at least a 
hundred times greater than those of the Taurus-cluster, it 
is just possible that the associated stars may be dispersing 
rather rapidly. 



4V 

I 

FIG. 4. Moving Cluster of " Orion " Stars in Perseus. 

A group to which the name moving cluster may be 
applied more legitimately is to be found in the constellation 
Perseus ; it was detected simultaneously by J. C. Kapteyn, 
B. Boss, and the writer. If we examine all the stars of 
the Orion type (Type B) in the region of the sky between 
K. A. 2 h and 6 h and Dec. + 36 and + 70 (about one-thirtieth 
of the whole sphere), we shall find that their motions fall 
into two groups. In Fig. 4 the motion of each star is 



iv MOVING CLUSTERS 65 

denoted by a cross, the star having a proper motion which 
would carry it from the origin to the cross in a century. 
If all the stars were to start from the origin at the same 
instant with their actual observed proper motions, then 
after the lapse of a century they would be distributed as 
shown in the diagram. Only one star has travelled 
beyond the limits of the figure and is not shown; with 
this exception the figure includes all the Type B stars in 
the region for which data are available. 

The upper group of crosses, which is close to the origin, 
consists of stars with very minute proper motions, all less 
than 1"*5 per century, and scarcely exceeding the probable 
error of the determinations. These are clearly the very 
remote stars, and there is not the slightest evidence that 
they are really associated with one another ; they appear 
to cling together because the great distance renders their 
diverse motions inappreciable. The lower group consists 
of seventeen stars sharing very nearly the same motion 
both as regards direction and magnitude. They evidently 
form a moving cluster similar in character to those we 

o 

have considered. Their association is further confirmed 
by the fact that'they are not scattered over the whole area 
investigated, but occupy a limited region of it. 

Table 7 shows the stars which constitute this group. It 
has been pointed out by T. W. Backhouse 5 that Nos. 742 
to 838 form part of a very striking cluster visible to the 
naked eye. The stars a Persei and o- Persei, which are 
not of the Orion Type, are included in the visual cluster ; 
the motion of the latter shows that it has no connection 
with this system, but a Persei appears to belong to it and 
may therefore be added to the group. The other stars in this 
part of the sky have also been examined so far as possible, 
but none of them show any evidence of connection with 
the moving cluster. All but three of the stars are 
arranged in a sort of chain, which may indicate a flat 
cluster (on the plan of the Ursa Major system) seen edge- 



66 



STELLAR MOVEMENTS 



CHAP. 



ways. It is always likely that some spurious members 
may be included through an accidental coincidence of 
motion, and it may be suspected that the three outlying 
stars are not really associated with the rest ; on the other 
hand, they may well be regarded as original members, 
which have been more disturbed by extraneous causes 
than the others. 

Owing to the small proper motion, the convergent point 
of this group cannot well be determined. The motion 
deviates appreciably from the direction of the solar antapex ; 
so that this cluster possesses some velocity of its own 
apart from that attributable to the Sun's own motion. 

TABLE 7. 
Mooing Cluster in Perseus. 



Boss's 
No. 


Name of Star. 


Type. 


Mag. 


R.A. 


Dec. 


Centen- 
nial 
Motion. 


Direc- 
tion. 










h. m. 





/> 


, o 


678 


Pi. 220 .... 


Bo 


5-6 


2 54 


+ 52 


4-3 


51 


740 


30Persei . . . 


B5 


5-5 


3 11 


4-44 


3-8 


55 


742 


29 Persei . . . 


B3 


5-3 


3 12 


+ 50 


4-5 


52 


744 


31 Persei . . . 


B3 


5-2 


3 12 


+ 50 


4-2 


51 


767 


Pi. 37 ... 


Bo 


5-4 


3 16 


+ 49 


3-6 


45 


780 


Brad. 476 ... 


B8 


5-1 


3 21 


+ 49 


3-2 


47 


783 


Pi. 56 


B5 


5-8 


3 22 


+ 50 


4*9 


37 


790 


34 Persei . . . 


B3 


4-8 


3 22 


+ 49 


4-4 


-* 

57 


796 


Brad. 480 ... 


B8 


6-1 


3 24 


+ 48 


4-7 


54 


817 


v/f Persei . . . 


B5 


4-4 


3 29 


+ 48 


4-3 


42 


838 


8 Persei .... 


B5 


3-0 


3 36 


+ 47 


4-6 


51 


898 


Pi. 186 .... 


B5 


5-5 


3 49 


+ 48 


3-9 


40 


910 


t Persei .... 


BO 


2-9 


3 51 


+ 40 


3-9 


49 


947 


c Persei .... 


B3 


4-2 


4 1 


+ 47 


4-4 


43 


1003 


d Persei .... 


B3 


4-9 


4 14 


+ 46 


4-5 


55 


1253 


15 Camelopardi . 


B3 


6'4 


5 11 


+ 58 


3-5 


37 


1274 


p Aurigae . . . 


B3 


5'3 


5 15 


+ 42 


4-5 


40 


772 


a Persei .... 


F5 


17 


3 17 


+ 50 


3-8 


55 



In the last column the "direction" is the angle between the direction of 
motion and the declination circle at 4 hours R.A. 

The tracing of these connections between stars widely 
separated from one another is an important branch of 



iv MOVING CLUSTERS 67 

modern stellar investigation ; and, as the proper motions 
of more stars become determined, it is likely that further 
interesting discoveries will be made. It seems worth while 
at this stage to consider what are the exact criteria by 
which we may determine whether a group of stars possesses 
that close mutual relation which is denoted by the term 
" moving cluster." Since some thousands of proper 
motions are available, it must be possible, if we take 
almost any star, to select a number of others the 
motions of which agree with its motion approximately. 
This is especially the case if, the parallax and radial 
velocity being unknown, the direction of motion is alone 
considered ; but, even if the velocities were known in all 
three co-ordinates, we could pick out groups which would 
agree approximately ; j ust as in a small volume of gas 
there must be many molecules having approximately 
identical velocities. Clearly the agreement of the motions 
is no proof of association, unless there is some further 
condition which indicates that the coincidence is in some 
way remarkable. There is a further difficulty that, as we 
have already seen in Chapter II. , stars, scattered through 
the whole region of the universe that has been studied, 
show common tendencies of motion, so that they have 
been divided into the two great star-streams. We must 
be careful not to mistake an agreement of motion arising 
from this general cosmical condition for the much more 
intimate association which is seen in the Taurus and Ursa 
Major systems. 

In the case of the Taurus and Perseus clusters the discrimi- 
nation is comparatively simple. These are compact groups 
of stars, so that only a small region of the sky and a small 
volume of space are considered, and the extraneous stars 
which might yield chance coincidences are not numerous. 
In the Taurus-cluster the large amount of the motion 
makes the group remarkable ; and, although in the 
Perseus-cluster the proper motion is not so great, we have 

F 2 



68 STELLAR MOVEMENTS CHAP. 

been careful to show by the diagram that it is very dis- 
tinctive. In the latter cluster, moreover, the resemblance 
of its members in type of spectrum helped to render the 
detection possible. 

The Ursa Major system, which is spread over a large 
part of the sky, presents greater difficulties. The dis- 
crimination of its more scattered members was only 
possible owing to the fact that its motion is in a very 
unusual direction. Its convergent point is a long way 
from the apex of either star-stream and from the solar 
apex, and stars moving in or near that direction are rare. 
In the writer's investigation of the two star-streams based 
on Boss's Preliminary General Catalogue, a striking 
peculiarity was presented in one region, which on enquiry 
proved to be due to five stars of this system ; that five 
stars moving in this way should attract attention, 
sufficiently illustrates the fact that motion in this par- 
ticular direction is exceptional. We are thus to a large 
extent safeguarded from chance coincidences ; never- 
theless, our ground is none too certain, and it may 
reasonably be suspected that one or two of the members 
at present assigned to the group will prove to be spurious. 

When the supposed cluster is not confined to one part 
of the sky and to one particular distance from the Sun, 
when there is nothing remarkable in its assigned motion, 
and when the choice of stars is not sufficiently limited, by 
the consideration of a particular spectral type or otherwise, 
not much weight can be attached to an approximate 
agreement of motion. A careful statistical study of groups 
in these adverse circumstances may eventually lead to 
important results ; but for the present we cannot be satis- 
fied to admit clusters the credentials of which do not 
reach the standard that has been laid down. 

In concluding this chapter we may try to sum up the 
importance of the discovery of moving clusters in stellar 
a-tromony. An immediate result is that in the Taurus 



iv MOVING CLUSTERS 69 

and Ursa Hajor stream we have been able to arrive at 
precise knowledge of the distance, relative distribution, 
and luminosity of stars which are far too remote for the 
ordinary methods of measurement to be successful. An im- 
portant extension of this knowledge may be expected when 
the proper motions of fainter stars have been accurately 
determined. Further, the possibility of stars widely 
sundered in space preserving, through their whole life-time 
up to now, motions which are equal and parallel to an 
astonishingly close approximation, is a fact which must be 
reckoned with when we come to consider the origin and 
vicissitudes of stellar motions. Generally the stars which 
show these associations are of early types of spectrum ; 
but in the Taurus cluster there are many members as far 
advanced in evolution as our Sun, some even of type K, 
whilst in the more widely diffused Ursa Major system 
there are three stars of type F. Some of these systems 
would thus appear to have existed for a time comparable 
with the life-time of an average star. They are wandering 
through a part of space in which are scattered stars not 
belonging to their system interlopers penetrating right 
among the cluster stars. Nevertheless, the equality of motion 
has not been seriously disturbed. It is scarcely possible 
to avoid the conclusion that the chance attractions of stars 
passing in the vicinity have no appreciable effect on stellar 
motions ; and that if the motions change in course of time 
(as it appears they must do) this change is due, not to the 
passage of individual stars, but to the central attraction 
of the whole stellar universe, which is sensibly constant 
over the volume of space occupied by a moving cluster. 

REFERENCES. CHAPTER IV. 

1. 'rYo/i//<</<'/i Publication*, No. 14, p. 87. 

-. (rfaninfien P>il>lii-<ifii>n*, No. 23. 

3. Plummer, Monthly X f '., Vol. 73. p. 4<>r>, Table X. (last two columns). 

4. Turner, The Observatory, Vol. 34, p. -J4r,. 

5. Backhouse, Monthly y<>(i<-<>.-<. Vol. 71, p. 523. 



70 STELLAR MOVEMENTS CH. iv 

BIBLIOGRAPHY. 

Taurus Cluster L. Boss, Astron. Joum., No. 604. 

Ursa Major Cluster .... Ludendorff, A sir. Nar,h., No. 4313-14; 

Hertzsprung, Astrophytu'id Joiim., Vol. 

30, p. 135 (Erratum, p. 320). 
Perseus Cluster Kapteyn, Internat. Solar Union, Vol. 3, 

p. 215 ; Benjamin Boss, Astron. Journ., 

No. 620 ; Eddington, Monthly Notices, 

Vol. 71, p. 43. 

There are in addition two less defined clusters, which have attracted some 
attention, viz., a large group of Type B stars in Scorpius and Centaurus, and 
the 61 Cygni group, which consists of stars scattered over most of the sky 
having a linear motion of the large amount 80 km. per sec. 5 

Scorpius-Centaurus Cluster . Kapteyn, loc. cit., p. 215; Eddington, loc. 

cit., p. 39. 
61 Cygni Cluster B. Boss, Astwn. Joum., Nos. 629, 633. 



CHAPTER V 

THE SOLAR MOTION 

IT was early recognised that the observed motions of the 
stars were changes of position relative to the Sun, and that 
part of the observed displacements might be attributed to 
the Sun itself being in motion. The question " What is 
the motion of the Sun ? " raises at once the philosophical 
difficulty that all motion is necessarily relative. In reality 
the manner in which the observed motion is to be divided 
between the Sun and the star is indeterminate ; these 
bodies are moving in a space absolutely devoid of fixed 
reference marks, and the choice and definition of a 
framework of reference that shall be considered at rest is 
a matter of convention. Probably philosophers of the 
last century believed that the undisturbed aether provided 
a standard of rest which might suitably be called absolute ; 
even if at the time it could not be apprehended in practice, 
it was an ultimate ideal which could be used to give 
theoretical precision to their statements and arguments. 
But according to modern views of the sether this is no 
longer allowable. Even if we do not go so far as to discard 
the aether-medium altogether, it is generally considered 
that no meaning can be attached to the idea of measuring 
motion relative to it ; it cannot be used even theoretically 
as a standard of rest. 

In practice the standard of rest has been the " mean of 

71 



72 STELLAR MOVEMENTS CHAP. 

the stars," a conception which may be difficult to define 
rigorously, but of which the general meaning is sufficiently 
obvious. Comparing the stars to a flock of birds, we can 
distinguish between the general motion of the flock and 
the motions of particular individuals. The convention is 
that the flock of stars as a whole is to be considered at 
rest. It is not necessary now to consider the reasons that 
may have suggested that the mean of the stars was an 
absolute standard of rest ; it is sufficient to regard it as a 
conventional standard, which has considerable usefulness. 
If there is any real unity in the stellar system, we may 
expect to obtain a simpler and clearer view of the pheno- 
mena by referring them to the centroid of the whole rather 
than to an arbitrary star like the Sun. By the centroid is 
meant in practice the centre of mass (or rather the centre 
of mean position) of those stars which occur in the 
catalogues of proper motions that are being discussed. 
As it is only the motion of this point that is being 
considered, its actual situation in space is not of conse- 
quence. If the motion of the centroid varied considerably 
according to the magnitude of the stars used or the 
particular region of the sky covered by the catalogue, it 
would be a very inconvenient standard. It is not yet 
certain what may be the extent of the variations arising 
from a particular selection of stars ; but, as the data of 
observation have improved, the wide variations shown in 
the earlier investigations have been much reduced or 
satisfactorily explained. At the present day, whilst few 
would assert that the " mean of the stars " is at all a 
precise standard, the indeterminateness does not seem 
sufficiently serious to cause much inconvenience. 

The determination of the motion of the stars in the 
mean relative to the Sun, and the determination of the 
solar motion (relative to the mean of the stars) are two 
aspects of the same problem. The relative motion, which- 
ever way it is regarded, is shown in our observations by a 



v THE SOLAR MOTION 73 

strong tendency of the stars to move towards a point in 
the sky, which according to the best determinations is near 
ft Columbae. Although individual stars may move in 
widely divergent or even opposite directions, the tendency 
is so marked that the mean of a very few stars is generally 
sufficient to exhibit it. Sir William Herschel's l first 
determination in 1783 was made from seven stars only, 
vt't lie was able to indicate a direction which was a good 
first approximation. From his time up till recent years 
the determination of the solar motion was the principal 
problem in all statistical investigations of the series of 
proper motions which were measured from time to time. 
This investigation was usually associated with a determina- 
tion of the constant of precession a fundamental quantity 
which is closely bound up with the solar motion in the 
analysis. In fact, both quantities are required to define 
our framework of reference ; the solar motion defines what 
is to be regarded as a fixed position, and the precession- 
constant defines fixed directions among the continually 
shifting stars. The numerous older determinations of the 
solar motion are now practically superseded by two results 
published in 1910-11, which rest on the best material yet 
available. 

The determination by Lewis Boss 2 from the proper 
motions of his Preliminary General Catalogue of 6188 
Stars gives, 

fR.A. 270-5 1-5 
SolarA P ex \Dec. +34-3 1-3 

The determination by W. W. Campbell 3 from the radial 
velocities (measured spectroscopically) of 1193 stars 
gives, 

fR.A. 268-5 2-0 
^olarApex \Dec. +25-3 1-8 

Speed of the solar motion 19'50'6 kilometres per second. 

The probable errors are not given by Campbell ; but the 



74 STELLAR MOVEMENTS CHAP. 

foregoing approximate values are easily deduced from the 
data in his paper. 

The discordance in declination between these two 
results, derived respectively from the transverse and the 
radial motions, is considerably greater than can be attributed 
to the accidental errors of the determinations. Possibly 
the discordance may be attributed to the different classes 
of stars used in the two investigations. Campbell's result 
depends almost wholly on stars brighter than 5 m *0, whereas 
Boss included all stars to the sixth magnitude and many 
fainter stars. Moreover, Boss's result depends more especially 
on the stars nearest to the Sun ; for in forming the mean 
proper motion in any region the near stars (having the 
largest angular motions) have most effect, whereas in 
forming the mean radial motion the stars contribute equally 
irrespective of distance. So far as can be judged, however, 
these differences will not explain the discordance. Boss 
made an additional determination of the solar apex, 
rejecting stars fainter than 6 m> ; the resulting position 
RA. 269'9, Dec. + 34'6 is almost identical with his 
main result. The writer, 4 examining the same proper 
motions on the two star-drift theory, by a method which 
gives equal weight to the near and distant stars, arrived at 
the position R.A. 267'3,' Dec. + 36'4, again a scarcely 
appreciable change. The cause of the difference between 
the results from the proper motions and the radial motions 
thus remains obscure. 

One of the most satisfactory features of Boss's deter- 
mination of the solar apex is the accordance shown by the 
stars of different galactic latitudes. If there is any 
relative motion between stars in different parts of the sky, 
it would be expected to appear in a division according to 
galactic latitude. The following comparison of the results 
derived from regions of high and low galactic latitudes is 
given by Boss. 



v THE SOLAR MOTION 75 

Solar Apex. 

Galactic Latitude of Zones. R.A. Dec. 

- 7 D to +7 D 269 40' +33 17' 

-19 to - 7 3 , and +19 to + 7 270 55 29 52 

-42 to -19, and +42 to +19 269 51 34 18 

S. Gal. Pole to -42, and N. Gal. Pole to +42 270 32 36 27 

The differences are quite as small as could be expected 
from the accidental errors. 

Another comparison of different areas of the sky can 
be made from the results of an analysis by the two-drift 
theory, which has the advantage that the position depends 
equally on all the stars used, instead of (as in the ordinary 
method) the nearest stars having a preponderating share. 

We find,- 

Solar Apex. 
Region. R.A. Dec. 

P*rA g*t5Sig} 265-5 + 37-0 

Equatorial Area Dec. -36 to + 36 3 269 '4 + 36'4 

There is thus a very satisfactory stability in the position 
of the apex determined from different parts of the sky.* 
The evidence is less certain as to its dependence on the 
magnitude and spectral type of the stars. There is some 
indication that the declination of the apex tends to increase 
for the fainter stars ; but it is not entirely conclusive. 
The range of magnitude in Boss's catalogue is scarcely grea't 
enough to provide much information ; so far as it goes, it 
is opposed to the view that there is any alteration in the 
apex for stars of different magnitudes, for, as already 
mentioned, the stars brighter than 6 m '0 give a result 
almost identical with that derived from the whole 
catalogue. From the Groombridge stars (Dec. +38 to 
N. Pole), F. \V. Dyson and W. G. Thackeray 5 found- 

* In the foregoing comparisons antipodal regions have always been taken 
together. There remains a possibility of a discordance between opposite 
hemispheres. 



76 STELLAR MOVEMENTS CHAP. 

Solar Apex. 

Magnitude. , s 

m. in. R.A. Dec. No. of Stars. 

1-0-4-9 245 3 +16-0 200 

5-0-5-9 268 + 27 = '0 454 

6-06-9 278 + 33-0 1,003 

7-0-7-9 280 +38^5 1,239 

8-08-9 272 3 +43-0 811 

This shows a steady increase in declination with diminishing' 
magnitude. It must, however, be noted that the area covered 
by the Groombridge Catalogue is particularly unfavourable 
for a determination of the declination of the apex. 

Other evidence pointing in the same direction has been 
found by G. C. Comstock, who made a determination of 
the solar motion from 149 stars of the ninth to twelfth 
magnitudes. He was able to obtain proper motions of 
these stars, because they had been measured micro- 
metrically as the fainter companions of double stars, but 
had been found to have no physical connection with the 
principal stars. The resulting position of the apex is 

R.A. 300 Dec. +54 

In a more recent investigation 7 the same writer has used 
479 faint stars, with the results 

Magnitude 7 m> to 10 m '0 Apex R.A. 280 Dec. +58 

10 m -0 13 m -0 R.A. 288 : Dec. +71 

The weight of these determinations cannot be great, but 
they tend to confirm the increase of declination with the 
faintness of the stars. 

Earlier investigations in which the stars were classified 
by magnitude are those of Stumpe and Newcomb. Tint 
former, using stars of large proper motion only, found a 
considerable progression in declination with faintness. 
Newcomb, on the other hand, who used stars of small 
proper motion only, found that the declination is steady. 
We now know that, owing to the phenomenon of star- 
streaming, the exclusion of stars above or below certain 
limits of motion is not legitimate, so that the contradictory 
character of these two results is not surprising. 



v THE SOLAR MOTION 77 

The comparative uncertainty of the proper motions of 
the fainter stars requires that results based on them 
should be received with caution. In particular, since the 
mean distance of the stars increases with faintness, the 
average parallactic motion becomes smaller, and a sys- 
tematic error in the declinations of any zone has a greater 
effect on its apparent direction. This is particularly 
serious, because these investigations have been usually 
based on northern stars only or on an even less extensive 



region. 



R.A. 


Dec. 


No. of Stars. 


274= -4 


+ 34 -9 


490 


270-0 


28-3 


1,647 


265 3 '9 


28-7 


656 


259-3 


42-3 


444 


275-4 


40-3 


1,227 


273-6 


38-8 


222 



There is fairly consistent evidence that the declination 
of the solar apex depends to some extent on the spectral 
type of the stars, being more northerly for the later types. 
In Boss's investigation 8 the following results were found, 

Solar Apex. 

Type. 

Oe5 B5 

B8 A4 

A5-F9 

G 

K 

M 

The later types G, K, M thus yield a declination differing 
markedly from the earlier types ; or, if we prefer to set 
aside the results for the groups containing few stars, which 
may be subject to large accidental errors, and confine 
attention to types A (B8 A4) and K, the difference of 
12 between the results of these two classes is evidently 
significant. 

The results of Dyson and Thackeray from the Groom- 
bridge stars show the same kind of progression. 

Solar Apex. 

Type. R'A. ~~De7. No. of Stars. 

B, A 269 +23^ 1100 

F, G, K 27M :;; 866 

Other investigations of this relation depend mainly on 
stars now included in Boss's catalogue, and used in his 



7 8 STELLAR MOVEMENTS CHAP. 

discussion. It therefore does not seem necessary to quote 
them. 

To sum up the results we have arrived at, it appears 
that we can assign a point in the sky at about R.A. 270, 
Dec. +34 towards which the motion of the Sun relative to 
the stellar system is directed. For some reason at present 
unknown, the determinations of this point by means of the 
spectroscopic radial velocities differ appreciably from those 
based on the transverse motions, giving a declination 
nearly 10 lower than the point mentioned. When 
different parts of the sky are examined the results are 
generally in good agreement, so that there can be little 
relative motion of the stars as a whole in different regions. 
There is some evidence that the solar apex increases in 
declination as successively fainter stars are considered, and 
it seems certain that for the later types of spectrum the 
declination is higher than for the earlier types. From all 
causes the solar apex from a special group of stars may 
(apart from accidental error) range from about +25 to 
+ 40 in declination ; variations in right ascension appear 
to be small and accidental. 

The speed with which the Sun moves in the direction 
thus found can only be measured from the radial motions. 
The result derived from the greatest amount of data is 
19 '5 km. per sec. 

Attention has been lavished on the investigation of the 
solar motion, not only on account of its intrinsic interest, 
but also because it is a unit of much importance in many 
investigations of the distribution of the more distant stars. 
The annual or centennial motion of the Sun is a natural 
unit of comparison in dealing with the stellar system, 
generally superseding the radius of the earth's orbit, which 
is too small to be employed except for a few of the nearest 
stars. It provides a far longer base-line than can be 
obtained in parallax-observations ; for the annual motion 
of the Sun amounts to four times the radius of the earth's 



v THE SOLAR MOTION 79 

orbit, and the motion of fifty or a hundred years, or even 
longer, may be used. The apparent displacement of the 
star attributable to the solar motion is called the 
parallactic motion. By determining the parallactic 
motion (in arc) of any class of stars their average 
distance can be found, just as the distance of an indi- 
vidual star is found from its annual parallax. It is not 
possible to find by observation the parallactic motion of an 
individual star, because it is combined with the star's 
individual motion ; but for a group of stars which has no 
systematic motion relative to the other stars, these indi- 
vidual motions will cancel in the mean. 

It may be appropriate to add some remarks on the 
theory of the determination of the solar motion from the 
observations. The method usually adopted for discussing 
a series of proper motions is that known as Airy's. 

Take rectangular axes, Ox being directed to the vernal 
equinox, Oy to R.A. 90, and Oz to the north pole. The 
parallactic motion (opposite to the solar motion) may be 
represented by a vector, with components X, Y, Z, directed 
to the solar antapex. X, Y, Z are supposed to be expressed 
in arc, so as to give the parallactic motion of a star at a 
distance corresponding to the mean parallax of all the stars 
considered. 

Taking a small area of the sky let the mean proper 
motion of the stars in the area be /* a , ^ in right ascension 
and declination respectively. Then considering the pro- 
jection of (X, F, Z) on the area considered, we have 

- X sin a + Y cos a = /u a 

X cos a sin 8 Y sin a sin d + Z cos 6 = pi 

where it is assumed that these stars are at the same mean 
distance as the rest, and that their individual motions 
cancel out. If these assumptions are not exactly satisfied, 
the deviations are likely to be mainly of an accidental 



: 



8o STELLAR MOVEMENTS CHAP. 

character. Taking the above equations for each region, a 
least-squares solution may then be made to determine 
X, Y, Z. The right ascension and declination A, D of the 
solar antapex are given by 

tan A = Y/X 
tanD = Z;(X*+Y*)l 

Additional terms in the equations of condition involving 
the correction to the preoessional constant and to the 
motion of the equinox are often inserted, but these need 
not concern us here. When stars distributed uniformly 
over the whole sky are considered, the additional terms have 
no effect on the result. 

Although the argument is clearer when we use the 
mean proper motion over an area for forming equations of 
condition, it is quite legitimate to use each star separately. 
For it is easily seen that the resulting normal equations 
are practically identical in the two procedures. In using 
the mean proper motion, it is easier and more natural to 
give equal weights to equal areas of the sky instead of 
weighting according to the number of stars ; this is 
generally an advantage. Further the numerical work is 
shortened. 

There are two weak points in Airy's method. First the 
mean proper motion (which, if not formed separately for 
each area, is virtually formed in the least-squares solution) 
is generally made up of a few large motions and a great 
number of extremely small ones. It is therefore a very 
fluctuating quantity, the presence or omission of one or 
two of the largest motions making a big difference in the 
mean. In a determination based nominally on 6000 stars, 
the majority may play only a passive part in the result, 
and the accuracy of the result is scarcely proportionate to 
the great amount of material used. The second point is 
more serious, since it leads to systematic error. We have 
, i umed that the mean parallax of the stars in each area 
differs only by accidental fluctuations from the mean of 






v THE SOLAR MOTION 81 

the whole sky ; but this is not the case. The stars near the 
galactic plane have a systematically smaller parallax than 
those near the galactic poles. 

It has often been recognised that this property of the 
galactic plane may cause a systematic error in the apex 
derived from the discussion of a limited part of the sky. 
Perhaps it is not so generally known that it will also cause 
error even when the whole sky is used. It seems worth 
while to examine this point at length. Happily it turns 
out that the error is not very large, but this could scarcely 
have been foreseen. 

If the mean parallax in any area is p times the average 
parallax for the whole sky, we may take account of the 
variation with galactic latitude by setting 

p = l + f P 2 (cos 6) 

where 6 is the distance from the galactic pole, e is a 
coefficient and 



We shall consider the case when the observations extend 
uniformly over the whole sky. 

The equations of condition should then read 

Xp sin a + Yp cos a = fi a 

Xp cos a sin 8 Yp sin a sin 8 + Zp cos 8 = /ig. 

We wish to re- interpret the results of an investigator who 
has not taken p into account. We therefore form normal 
equations, just as he would do, viz., from the right 
ascensions : 

X'S.p sin -a - Y^p sin a cos a = - S^i a sin a 

X"S,p sin a cos a + YZp cos -a = S/j a cos a. 

and from the declinations : 



X^p cos -'a sin -8 4- YZp sin a cos a sin -8 Z'S.p cos a sin 5 cos 8 = 

- 2^5 cos a sin 8 
X2p sin a cos a sin 2 8 + Y^.p sin 2 a sin -8 Z2p sin a sin 8 cos 8= 

2^s sin a sin 8 

- A'2p cos a sin 8 cos 8 - Y2p sin a sin 8 cos 8 + ZXp cos -8 = i>6 cos 8 

G 



82 STELLAR MOVEMENTS CHAP. 

giving the combined equations : 

X'Zp (sin 2 a + cos -a sin 2 8) Y2p sin a cos a cos 2 8 Z'S.p cos a sin 8 cos 8 = 

- 2(/u a sin a + ^5 cos a sin 8) 

X"2.p sin a cos a cos 2 8 + Y2p (cos 2 a + sin -a cos 2 8) Z'S.p sin a sin 8 cos 8 = 

2(/i a cos a ps sin a sin 8) 
- X'S.p cos a sin 8 cos 8 - YZp sin a sin 8 cos 8 + Z^.p cos -8 = 2/xg cos 8. 

Now it clearly can make no difference in a least-squares 
solution whether we resolve our proper motions in right 
ascension and declination or in galactic latitude and 
longitude. The value of the solar motion, which makes 
the sum of the squares of the residuals in R.A. and Dec. 
a minimum, must be the same as that which makes the 
sum of the squares of the residuals in Gal. Lat. and Long. 
a minimum. We may therefore treat a solution as though 
it had been made in galactic co-ordinates, although the 
actual work was done in equatorial co-ordinates. 

Let then a, S now stand for galactic longitude and 
latitude, so that A", F, Z is the paral lactic motion vector 
referred to rectangular galactic co-ordinates. We shall 
have 



Taken over a whole sphere the mean value of 



2 1 

p(sin -a + cos 2 a sin 2 8) = - + ~e 

3 15 



3 lo 

cos 2 8 = - -_? e 

3 15 



The other coefficients vanish when integrated over a 
sphere. Thus the normal equations become (setting N for 
the total number of stars used) 

2 / 1 \ 

x JT ( 1 + J~Q J = - 2 (pa sin a + (i& cos a sin 8)-r- N 

2 r / 1 \ 

o Y ( 1 -f jTj J = 2 (/* a cos a ps sin u sin 8) -r- N 



v THE SOLAR MOTION 83 

And if X , y o , ZQ are the solutions obtained when the P 2 
term is neglected, 



The original and corrected galactic latitudes of the antapex 
bein \ , X, we have 



tan X = 



l-t< 



whilst the galactic longitude is unaltered. 

The effect of the decrease of parallax towards the galactic 
plane is thus to make X numerically less than \. The 
uncorrected position of the solar apex is too near the 
galactic plane. 

Inserting numerical values, X = 20, and e may perhaps 
be ^ (i.e., mean parallax at the pole / mean parallax in the 
plane =8/5), we find X = 2157 r . The correction is 
just under 2. Reverting to equatorial co-ordinates, the 
correction is mainly in right ascension, the right ascension 
given by the ordinary solution being about 2'4 too great. 

It is quite practicable to work out the corresponding 
corrections, when the proper motions cover only a zone of 
the sky limited by declination circles. In this case we 
have to retain equatorial co-ordinates throughout, and 
express P 2 (cos 6) in terms of a and S. The mean values 
of the functions of sin a, cos a, sin S and cos S that occur 
are readily evaluated for the portion of the sphere used. 
As the numerical work depends on. the particular zone 
chosen, we shall not pursue this matter further. 

A second method of finding the solar apex from the 
proper motions, known as Besse.l's method, has been used 
by H. Kobold. 9 Each star is observed to be moving along 
a great circle on the celestial sphere. Consider the poles 
of these great circles. If the stellar motions all converged 
to a point on the sphere, the poles would all lie along the 

G 2 



84 STELLAR MOVEMENTS CHAP. 

great circle equatorial to that point. Thus a tendency of 
the stars to move towards the solar antapex should be 
indicated by a crowding of the poles towards the great 
circle equatorial to the antapex. This affords a means of 
finding the direction of the solar motion by determining 
the plane of greatest concentration of the poles. . It is to 
be noted, however, that this method makes no discrimina- 
tion between the two ways in which a star may move along 
its great circle. Two stars moving in exactly opposite 
directions will have the same pole. A paradoxer might 
argue that, as the effect of the solar motion is to cause a 
minimum number of stars to move towards the solar apex, 
the solar motion will be indicated by a tendency of the 
poles to avoid the plane equatorial to its direction. How- 
ever, if the individual motions are distributed according to 
the law of errors, the crowding to the plane will be found to 
outweigh the avoidance, so that the method is legitimate 

o o 

though perhaps a little insensitive. But if the individual 
motions follow some other law, the result may be altogether 
incorrect. In the light of modern knowledge of the 
presence of two star-streams, Bessel's method can no 
longer be regarded as an admissible way of finding the 
solar apex ; but it is interesting historically, for in 
Kobold's hands it first foreshadowed the existence of the 
peculiar distribution of stellar motions which is the 
subject of the next chapter. 

The determination of the solar motion from the radial 
velocities presents no difficulty. If (X, Y, Z] is the vector 
representing the parallactic motion in linear measure, each 
Mar yields an equation of condition : 

X cos a cos d + Fsinacos5-H^sin^ = radial velocity. 

A least-squares solution is then made, the individual motions 
of the star> IM-MI- treated as though they were accidental 
errors. The numerical work can be shortened by using in 
the equations of Condition tin- nu-an radial velocity for a 



v THE SOLAR MOTION 85 

sniiill area of the sky, instead of the individual results. 
The resulting normal equations are practically unaltered, 
and there is no theoretical advantage. 

REFERENCES. CHAPTER V. 

1. Sir W. Herschel, Collected Papers, Vol. 1, p. 108. 

2. Boss, Ash-on. Journ., Nos. 612, 614. 

3. Campbell, Lick Bulletin, No. 196. 

4. Ecldingtoii, Monthly AW/<v.s, Vol. 71, p. 4. 

5. Dyson and Thackeray, Monthly Notices, Vol. 65, p. 428. 

6. Comstock, Astron. Journ., No. 591. 

7. Comstock, Astron. Journ., No. 655. 

8. Boss, Astron. Journ., Nos. 623-4. 

9. Kobold, Nova Ada der Kais. Leop. Carol. Deutschen Akad., Vol. 64 ; 
Atr. Nach., Nos. 3163, 3435, 3591. 

BIBLIOGRAPHY. 

The following references are additional to those quoted in the Chapter. 
Owing to the recent improvement in the data of observation, and the change 
of theoretical views due to the recognition of star-streaming, the interest 
of these papers is now, perhaps, mainly historical. 

Argelander, Memoires presenter a VAcad. des. Sci., Paris, Vol. 3, p. 590 

(1837). 

Bravais, Liouvilles's Journal, Vol. 8 (1843). 
Airy, Memoirs R.A.S., Vol. 28, p. 143 (1859). 
Stumpe, Astr. Nach., No. 3000. 
Porter, Cincinatti Trans., No. 12. 
L. Struve, Memoires St. Petersboury, Vol. 35, No. 3 ; Astr. Xach. Nos. 3729, 

3816. 

Newcomb, Astron. Papers of the American Ephemeri*. Vol. 8, Pt. 1. 
Kapteyn, Astr. Nach., Nos. 3721, 3800, 3859. 
Boss, Astron. Journal, No. 501. 
Weersma. Qroningen Publications, No. 21. 



CHAPTER VI 

THE TWO STAR STREAMS 

THE observed motion of any star can be regarded as 
compounded of two parts ; one part, which is attributable 
to the motion of the Sun as point of- reference, is the 
parallactic motion ; and the other residual part is the 
star's motus peculiaris or individual motion. It must 
be borne in mind that this division cannot generally be 
effected in practice for the proper motion of a star ; 
because, although the parallactic motion in linear measure 
is known, we cannot tell how much it will amount to in 
angular measure, unless we know the star's distance, 
and this is very rarely the case. On the other hand, the 
spectroscopic radial velocities, being in linear measure, can 
always be freed from the parallactic motion, if desired. 
As the greater part of our knowledge of stellar movements 
is derived from the proper motions, we cannot study the 
mottis peculiares directly, but must deduce the phenomena 
respecting them from a statistical study of the whole 
motions. 

In researches on the solar motion, it has usually, though 
not always, been assumed that the mottis peculiares of the 
stars are at random. This was the natural hypothesis 
to make, when nothing was known as to the distribution 
of these residual motions ; and certainly, when we consider 



N 



CH. vi THE TWO STAR STREAMS 87 

how vast are the spaces which isolate one star from 
its neighbour, and how feeble must be any gravitational 
forces exerted across such distances, it might well seem 
improbable that any general tendency or relation could 
connect the individual motion of one star with another. 
Yet many years ago the phenomenon of local drifts of stars, 
or, as they are now called, Moving Clusters, was known. 
But although such instances of departure from the strict 
law of random distribution of motions must have been 
recognised as occurring exceptionally, probably few 
astronomers doubted that the hypothesis was substantially 
correct. In 1904, however, Prof. J. C. Kapteyn l showed 
that there is a fundamental peculiarity in the stellar 
motions, and that they are not even approximately 
haphazard. This deviation is not confined to certain 
localities, but prevails throughout the heavens, wherever 
statistics of motions are available to test it. 

Instead of moving indiscriminately in all directions, as 
a random distribution implies, the stars tend to move in 
two favoured directions. It does not matter whether the 
parallactic motion is eliminated or not. A tendency to 
move in one favoured direction would disappear, when 
the parallactic motion was removed ; but a tendency in 
two directions can only be an intrinsic property of the 
individual motions of the stars. It may seem strange 
that this striking phenomenon was so long overlooked by 
those who were working on proper motions ; but usually 
investigators, having the solar motion mainly in their 
minds, as a first step towards gathering their data into 
a manageable form grouped together the stars in small 
regions of the sky, and used the mean motion. This 
unfortunately tends to conceal any peculiarity in the 
individual motions. In order to exhibit the phenomenon 
it is necessary to find some means of showing the statistics 
of the separate stellar motions ; this may be done conveni- 
ently in the following way. 



88 



STELLAR MOVEMENTS 



CHAP. 



ANALYSIS OF PROPER MOTIONS 

Confining our attention to a limited area of the sky, so 
that the apparent motions are seen projected on what is 
practically a plane, we count up the number of stars 
observed to be moving in the different directions. If in 
classifying directions we proceed by steps of 10, we shall 
then form a table of the number of stars moving in 36 
directions, towards position angles 0, 10, 20 . . . 350. 




Velocity 0-3 Unit 





Velocity 0-6 Unit 



Velocity 1-5 Unit 



FIG. 5. Simple Drift Curves. 

The result can be conveniently shown on a polar diagram, 
i.e., a curve is drawn so that the radius is proportional to 
the number of stars moving in the corresponding direction. 
Before considering the diagrams actually derived from 
observation, let us examine what form of curve would be 
obtained if the hypothesis of random motions were correct. 
The curve would not be a circle because of the parallactic 
motion ; for, as these observed motions are referred to the 
Sun, there would IM superposed on the random individual 
motions the motion of the star-swarm as a whole. If, for 
example, this latter motion were towards the north, then 



vi THE TWO STAR STREAMS 89 

clearly there would be a maximum number of stars moving 
north and fewest south, the number falling off symmetri- 
cally on either side from north to south. The exact form 
can be calculated on the hypothesis of random distribution ; 
it varies with the magnitude of the parallactic motion 
compared with the average motus peculiaris, being more 
elongated the greater the parallactic motion. In Fig. 5 

270 
240 300 



ISO- 




ISO 30 

12tf I 60 



90 

FIG. 6. Observed Distribution of Proper Motions. 
(Groombridge Catalogue II. A. 14 h to 18 h , Dec. +38' to +70'.) 

examples of this curve are given ; it may be noted how 
sensitive is the form of the curve to a small change in the 
parallactic velocity. It is convenient to have a name for a 
system such as is represented in these figures, in which 
the individual motions are haphazard, but the system as a 
whole is in motion relative to the Sun ; we call such a 
system a drift. 

As an example of a curve representing the observed 
distribution of proper motions we take Fig. 6. 2 This 



9 o STELLAR MOVEMENTS CHAP. 

corresponds to a region of the sky between K.A. 14 h and 
18 h , Dec. 4- 38 and 4- 70, the proper motions being taken 
from Dyson and Thackeray's " New Reduction of Groom- 
bridge's Catalogue." The motions of 425 stars are here 
summarised. It is quite clear that none of the single 
drift curves of Fig. 5 can be made to fit this curve 
derived from observation. Its form (having regard to 
the position of the origin) is altogether different. No one 
of the theoretical curves corresponds to it in even the 
roughest manner. It will be noticed that there are two 
favoured directions of motion ; the stars are streaming in 
directions 80aud 225, the latter being the more pronounced 
elongation. The actual number of stars moving in each 
direction is given in the fifth column of Table 8 below. 
Neither of the favoured directions coincides with that 
towards the solar antapex, viz., 205, for this part of the 
sky. It is true the mean motion of all these stars is 
towards the antapex, but we see that that is merely a 
mathematical average between the two partially opposed 
streams that are revealed in the diagram. 

It is possible to obtain a theoretical figure that will 
correspond approximately with Fig. 6 in the following 
way. Suppose, instead of the single drift we have 
hitherto considered, there are two star-drifts. Let one, 
consisting of 202 stars, be moving in the direction 225 
with velocity* 1*20, and the other, consisting of 232 stars, 
be moving in the direction 80 with the much smaller 
velocity 0*45. The corresponding curves are P and Q, 
Fig. 7. If these were seen mingled together in the sky, 
the resulting distribution would be represented by the 
curve R. Each radius of R is, of course, formed by 
adding together the corresponding radii of P and Q. If 
R is carefully compared with the observed curve, it will 
be seen that the resemblance is close. The numerical 

* The unit velocity l/7i, which is related to the mean individual motion, is 
denned in the mathematical theory in the next chapter. 



VI 



THE TWO STAR STREAMS 



comparisons, which these diagrams illustrate, are given in 
Table 8 ; it is there shown that by adding together two 

TABLE 8. 

Analysis of Proper Mat inns in the Region R.A. 14* to 18 A , 
Dec. +38 to +70. 





Calculated. 




Direction. 






Observed. 


Difference. 
Obs. Calc. 








Drift I. 


Drift II. 


Total. 








5 


066 


4 


-2 


15 


077 


5 -2 


25 


088 


6 


-2 


35 


10 10 


9 -1 


45 


11 11 


10 -1 


55 


12 12 


14 +2 


65 


12 12 


14 +2 


75 


13 13 


14 +1 


85 


13 13 


13 


95 


12 12 12 


105 


1 12 13 


10 -3 


115 


1 11 12 


11 -1 


125 


1 


10 11 


10 -1 


135 


1 


8 


9 


10 +1 


145 


2 


7 


9 


7 -2 


155 


3 


6 


990 


165 


5 


6 


11 


9 -2 


175 


7 


5 


12 


14 +2 


185 


11 


4 


15 14 - 1 


195 


15 


4 


19 16 -3 


205 


19 


3 


22 21 


-1 


215 


23 


3 


26 27 +1 


225 


24 


3 


27 29 


+ 2 


235 


23 


3 


26 26 





245 


19 


3 


22 19 


-3 


255 


15 


3 


18 17 


-1 


265 


11 3 


14 


12 


-2 


275 


7 3 


10 11 


+ 1 


285 


5 3 8 11 


+ 3 


295 


3 3 


6 8 


+ 2 


305 


2 3 


5 7 


+ 2 


315 


1346 


+ 2 


325 


1 4 


5 6 


+ 1 


335 


1 4 


5 5 





345 


1 


5 


6 5 


-1 


355 


6 


6 


4 


-2 


Totals . 


. 202 


232 


434 425 






92 STELLAR MOVEMENTS CHAP. 

theoretical drifts, the observed distribution of motions is 
approximately obtained. Without pressing the conclusion 
that a combination of two simple star-drifts will represent 
the actual distribution in all its detail, we may at least 
assert that it represents its main features, whereas not 
even the roughest approximation can be obtained with 





Q 

FIG. 7. Calculated Distributions of Proper Motions. 

a single drift, i.e., with the hypothesis of random- 
motions. 

As another illustration, we may take Fig. 8, which refers 
to a different part of the sky, the proper motions this time 
being taken from Boss's Preliminary General Catalogue. 
The uppermost curve, which has so interesting an appear- 
ance, is derived from the observed proper motions. Curve 
B is the best approximation that can be found on the 
assumption of a random distribution of motions plus 
the parallactic motion. It may be remarked that since 
the solar apex is a rather well determined point, the 
direction of elongation of the curve B is not arbitrary ; it 
is necessary to draw it pointing in the known direction of 
the parallactic motion. The curve C is an approximation 
by a combination of two star-drifts ; these again were not 
taken as arbitrary in direction, but were made to point 
towards appropriate apices deduced from a general discus- 
sion of the whole sky. It is quite probaldr that there are 
differences between A and C which are not purely 



VI 



THE TWO STAR STREAMS 



93 



accidental, but it will 
at least be admitted 
that, whereas the curve 
B bears scarcely any 
resemblance to the ob- 
served curve, the curve 
C - reproduces all the 
main features of the 
distribution, and from 
it we can if we choose 
proceed to investigate 
the irregularities of de- 
tail. 

The foregoing examples 
illustrate a method of 
analysis which has been 
applied successfully to a 
great many parts of the 
sky. It consists in find- 
ing, generally by trial 
and error, a combination 
of two drifts which will 
give a distribution of 
motions agreeing closely 

with that actually observed. In comparing the results 
obtained from different parts of the sky, it must 
be remembered that we are studying the two- 
dimensional projections of a three-dimensional phenom- 
enon, and the diagrams will vary in appearance as 
the circumstances of projection vary. The most accu- 
rate series of proper motions at present available 
is contained in Lewis Boss's " Preliminary General 
Catalogue," and it is of special interest to examine fully 
the results derivable from it. 3 The catalogue contains 
6188 stars well-distributed over the whole sky; practically 




Double-drift approximation 



FIG. 8. Observed and Calculated 
Distributions of Proper Motions. 
(Boss, Region VIII.) 



94 STELLAR MOVEMENTS CHAP. 

all stars down to the sixth magnitude are included, and the 
fainter stars appear to be fairly representative and have 
not been selected on account of the size of their proper 
motions. A very high standard has been attained in the 
elimination of systematic error the main cause of trouble 
in these researches though no doubt there is still a 
possibility of improvement in this respect. There can be 
no doubt that the catalogue represents the best data that 
it is at present possible to use. 

After excluding certain classes* of stars for various 
reasons, 5322 remained for investigation. These were 
divided between seventeen regions of the sky, each region 
consisting of a compact patch in the northern hemis- 
phere together with an antipodal patch in the southern 
hemisphere. By taking opposite areas together in this 
way we double the number of stars without unduly 
extending the region, for the circumstances of projection 
in opposite regions are identical. The regions were 
numbered from I. to XV IT., I. being the circular area 
Dec. 4- 70 to the Pole; II. to VII. formed the belt 
between + 36 = and +70 with centres at O 1 ', 4 h , 8 h , 12 h , 
16 h , and 20 h respectively; and VIII. to XVII. formed the 
belt to +36 with centres at l h 12 ra , 3 h 36'", etc. 
(These are the positions of the northern portions ; the 
antipodal part of the sky is also to be included in each 
case.) 

The diagrams for 11 of the 17 regions are given in 
Fig. 9. The arrows marked Antapex point to the antapex 
of the solar motion (R.A. 90*5, Dec. 34 *3) ; the arrows 
I. and II. point to the apices of the two drifts, found from 
the collected results of this discussion. It will be seen that 
the evidence for the existence of two star-streams is very 
strong. The tendency to move in the directions of the 
arrows I. and II. is plainly visible, and it is scarcely 

* Viz., stars of the Orion type, members of moving clusters, and the fainter 
components of binary systems. 



VI 



THE TWO STAR STREAMS 



95 



necessary again to emphasise that there is no resemblance 
to a symmetrical single-drift curve pointing along the 
antapex arrow. In certain cases, notably Regions XIV. 



270" 




i. Region I. ; centre, North Pole. 
270 



180 




ii. Region II. ; centre, R.A. O h , Dec. +50 3 . 

270 



180 




II 



in. Region V. ; centre, R.A. 12 h , Dec. + 50. 

FIG. 9. Diagrams for the Proper Motions of Boss's " Preliminary 
General Catalogue." 




STELLAR MOVEMENTS 

270 270 



180 



180 




iv. Region VI. ; centre R.A. 16 h , 
Dec. +50. 



270 



\ 



180 -, 



v. Region VII. ; centre R.A. 20 1 ', 
Dec. +50. 




II 



A 

Antsipex 
vi. Region VIII. ; centre, R.A. l h 12 m , Dec. +17. 



270 



180 




II 



90 



vii. Region XII. ; centre, R.A. 10 h 48 m , Dec. +17 . 



Fit;. 9 (continued). Diagrams for the Proper Motions of Boss' 
"Preliminary 



THE TWO STAR STREAMS 




II 

viii. Region XIII. ; centre R.A. 13 h 12 m , Dec. +17. 

270 



180 




90 II 

ix. Region XIV. ; centre R.A. 15 h 36 m , Dec. +17. 



180 



>- 




x. Region XVI. ; centre R.A. 20 h 24 m , Dec. +17 



FIG. 9 (continued) . Diagrams for the Proper Motions of Boss's 
"Preliminary General Catalogue." 

H 



9 8 



STELLAR MOVEMENTS 

270 



CHAP, 



180 




xi. Region XVII. ; centre R.A. 22 h 48 m , Dec. +17. 

FIG. 9 (continued). Diagrams for the Proper Motions of Boss's 
" Preliminary General Catalogue/' 

and XVI. , there appears to be a streaming towards the 
antapex in addition to the streaming in the directions of 
the two drifts, so that the -curve appears three-lobed like 
a clover leaf. This is an important qualification of our 
conclusion, but for the present we shall not discuss it ; 
later it will be considered fully. The eleven regions 
chosen for representation are those in which the separ- 
ation into two drifts ought to be most plainly indicated. 
It will be understood that there are parts of the sky in 
which the projection is such that they are not very plainly 
separated. In fact there must be one plane of projection 
on which the drifts have identical transverse motions, and 
would therefore become indistinguishable, except by 
having recourse to the radial velocities. The fact, then, 
that in the regions which are not here represented the 
phenomenon is shown less plainly, in no way weakens the 
argument, but rather confirms it. 

Let us suppose now that w r e have succeeded in analysing 
the stellar motions in each of the seventeen regions into 
their constituent drifts, and have thus determined the 
directions and velocities of the two drifts at seventeen 
points of the celestial sphere. If the drift motion in each 
region is really the same motion seen in varying projec- 
tions, we must find that <>n plotting the directions on a 



VI 



THE TWO STAR STREAMS 



99 



globe they will all converge to one point. This will be true 
for each drift separately. The convergence actually found is 
shown in Figs. 10 and 11. Imagine the great circles traced 



+ 30 



+20- 




+ 10- 



o 5 10 15 20' 

Approximate Scale 

FIG. 10. Convergence of the Directions of Drift I. from the 17 Regions. 



+65 




+50 



+55 



+45 



5 1 15 

Approximate Scale 



FIG. 11. Convergence of the Directions of Drift II. from the 17 Regions. 

on the sky and a photograph taken of the part of the sky 
where they converge ; the great circles on such a photograph 
will appear as straight lines. These are shown on the 

H 2 



ioo STELLAR MOVEMENTS CHAP. 

two diagrams, and the Roman numeral attached to each 
line indicates the region from which it comes. Each 
diagram represents an area of the sky measuring about 60 
by 30 ; this would correspond on the terrestrial globe to 
a map of Northern Africa from the Congo to the Mediter- 
ranean. The apex marked on each diagram is the defini- 
tive apex of the drift, determined by a mathematical 
solution. For one of the drifts, called Drift L, the con- 
vergence of the directions is so evident as to need no 
comment. Owing to the smaller velocity of Drift II., its 
direction in any region cannot be determined so accu- 
rately, and a greater deviation of the great circles must be 
expected. Having regard to this, the agreement must be 
considered good, Region VII. being the only one showing 
important discordance. To appreciate the evidence of 
this diagram we may make a terrestrial comparison ; if 
from seventeen points distributed uniformly over the 
earth tracks (great circles) were drawn, every one of which 
passed across the Sahara, they might fairly be considered 
to show strong evidence of convergence ; the distribu- 
tion of the Drift II. directions is quite analogous. 

The analysis of the regions gives not only the directions 
of the two drifts, but also their speeds in terms of a 
certain unit ; and both sets of results may be used in 
finding definitive positions of the apices towards which 
the two drifts are moving. The results of a solution by 
least squares are as follows : 

Drift I. Drift II. 



Apex. 


Apex. 


R.A. 


Dec. 


Speed. 


R.A 


Dec. 


Speed. 


10 Equatorial Regions 


( .>2 


4 


-14 


1 


1 


507 


286 


5 


-63 


6 


0-869 


7 Polar Regions . . 


HU- 


3 


- 1<> 


7 


1 


536 


289 


1 


83 


r> 


O'16 


Whole Sphere . . 


GO' 


8 


-14 C 


6 


1 


516 


287 


8 


-64 


1* 


1 0-855 



* The fact that the declination derived from tho whole sphere does not lie 
between the declinations from the two portions looks paradoxical, but is due 
to the unequal weights of the determinations of the Z component from the 
two portions. 



vi THE TWO STAR STREAMS >oi 

The speeds are measured in terms of the usual theo- 
retical unit 1 //. 

The drift-speed in an} 7 region should (owing to fore- 
shortening) vary as the sine of the angular distance from 
the drift-apex, being greatest 90 away from the apex 
and diminishing to zero at the apex and antapex. This 
progressive decrease as the regions get nearer the apex is 
well shown in the observed values, and the sine-law is 
followed with very fair accuracy. 

Another fact derived from the analysis is the proportion 
in which the stars are divided between the two streams ; 
this seems to vary somewhat from region to region, but 
the mean result is that 59*6 per cent, belong to Drift 
I. and 40*4 per cent, to Drift II. ; that is a proportion of 
practically 3:2. 

To sum up, the result of this analysis of the Boss 
proper motions is to show that the motions can be closely 
represented if there are two drifts. That which we have 
called Drift I. moves with a speed of 1'52 units, the other, 
Drift II., with a speed of 0'86 unit. The first drift 
contains f of the stars and the second drift f . Their direc- 
tions are inclined at an angle of 100. 

It will be remembered that these motions are measured 
relative to the Sun. In Fig. 12, let SA and SB repre- 

B c A 




FIG. 12. 



sent the drift-velocities, making an angle of 100. Divide 
AB at C so that A C : CB = 2:3 corresponding to the 
proportion of stars in the two drifts. Then SO represents 



STELLAR MOVEMENTS CHAP. 



the motion of the centroid of all the stars relative to the 
Sun, and accordingly CS represents the solar motion 
and points towards the solar apex. AB and BA repre- 
sent the motion of one drift relative to the other ; the 
points in the sky towards which this line is directed are 
called the Vertices. The positions found from the num- 
bers above are 



v fR.A. 94>-2 Dec. 

3 ' ' \R.A. 274=-2 Dec. -11 '9. 

The relative velocity of the two drifts is 1*87 units. 

It is a remarkable fact that the vertices fall exactly 
in the galactic plane, so that the relative motion of the 
two drifts is exactly parallel to the galactic plane. 

The solar motion CS found from the same numbers is 
0*91 towards the 

Solar Apex. . . R.A. 267 C '3 Dec. +36 '4 

This may be compared with Prof. Boss's determination 
from the same catalogue by the ordinary method 4 : 

Solar Apex. . . R.A. 270'5 Dec. + 34-3 

The agreement is interesting because the principles of 
the two determinations are very different ; moreover, in 
Boss's result the magnitudes of the proper motions as 
well as their directions were used, whereas the analysis 
on the two-drift theory depends solely on the directions. 

Since the speed of the solar motion has also been 
measured in kilometres per second, this provides an 
equation for converting our theoretical unit into linear 
measure. We have 0*91 unit =19*5 km./sec., whence 
the theoretical unit l/h is 21 kilometres per second. We 
can thus, if we wish, convert any of the velocities 
previously given into kilometres per second. 

It will b(- seen from Fig. 12 that the direction of motion 
of Drift I., SA, is inclined at a comparatively small angle 
to the parallactic motion, SC. But that the directions 



vr THE TWO STAR STREAMS 103 

are clearly distinct may be appreciated by referring back 
to Fig. 10. The solar an tapex actually falls just outside 
that diagram, so it is clear that the convergence is not 
towards the solar antapex but towards a different apex at 
the point indicated. 

When referred to the centroid of the stars instead of to 
the Sun, the motions CA and CB of the two drifts are 
seen to be opposite to one another. It is perhaps not 
easy to realise that the inclination of the two stream- 
motions is a purely relative phenomenon depending on 
the point of reference chosen ; but this is the case. If we 
divest our minds of all standards of rest and contemplate 
simply two objects in space two star-systems all that can 
be said is that they are moving towards or away from or 
through one another along a certain line. The distinction 
between meeting directly or obliquely disappears. It is 
clear that this line joining the vertices must be a very 
important and fundamental axis in the distribution of 
stellar motions. It is an axis of symmetry, along which 
there is a strong tendency for the stars to stream in one 
direction or the other. It is this point of view T that has 
led to an alternative mode of representing the phenomenon 
of star-streaming, the ellipsoidal theory of K. Schwarz- 
schild. 5 

We have, hitherto, analysed the stars into two separate 
systems, which move, one in one direction, the other 
in the opposite direction, along the line of symmetry ; 
but Schwarzschild has pointed out that this separation is 
not essential in accounting for the observed motions. It 
is sufficient to suppose that there is a greater mobility 
of the stars in directions parallel to this axis than in 
the perpendicular directions. The distinction is a little 
elusive, when it is looked into closely. It may be illus- 
trated by an analogy. Consider the ships on a river. 
One observer states that there are two systems of ships 
moving in opposite directions, namely, those homeward 



io 4 STELLAR MOVEMENTS CHAP. 

bound and those outward bound ; another observer makes 
the non-committal statement that the ships move 
generally along the stream (up or down) rather than 
across it. This is a not unfair parallel to the points 
of view of the two-drift and ellipsoidal hypotheses. The 
distinction is a small one and it is found that the two 
hypotheses express very nearly the same law of stellar 
velocities ; but by the aid of different mathematical 
functions. This will be shown more fully in the mathe- 
matical discussion in the next chapter. Meanwhile we 
may sum up in the words of F. W. Dyson 6 : " The 
dual character of Kapteyn's system should not be unduly 
emphasised. Division of the stars into two groups was 
incidental to the analysis employed, but the essential 
result was the increase in the peculiar velocities of stars 
towards one special direction and its opposite. It is this 
same feature, and not the spheroidal character of the 
distribution, which is the essential of Schwarzschild's 
representation." 

The phenomenon of star-streams (by which we mean the 
tendency to stream in two favoured directions, which both 
the two-drift and ellipsoidal theories agree in admitting) 
is shown very definitely in all the collections of proper 
motions that are available for discussion. Very careful 
attention has been given to the question whether it could 
possibly be spurious, and due to unsuspected systematic 
errors in the measured motions. 7 It is not difficult for the 
investigate* to satisfy himself that such an explanation is 
quite out of the question, but it is not so easy to give the 
evidence in a compact form. Happily we are able to 
give one piece of evidence which seems absolutely con- 
clusive. F. W. Dyson 8 has made an investigation of the 
stars (1924 in number) with proper motions exceeding 20" 
a century. In this case we are not dealing with small 
quantities just perceptible with refined measurements, but 



VI 



THE TWO STAR STREAMS 



105 



with large movements easily distinguished and cheeked. 
These large motions show the same phenomenon that has 
been described for the smaller motions of Boss's catalogue. 
In fact, the two streams are shown more prominently when 
we leave out the smaller motions. This does not mean 
that the more distant stars are less affected by star- 
streaming than the near stars ; it is easy to show that for 
stars at a constant distance the small proper motions must 

N 



Scale 
20" 30" 40" 



FIG. 13. Distribution of Large Proper Motions (Dyson). 

necessarily be distributed more uniformly in position angle 
than the large motions, and the enhancement of the stream- 
ing when the small motions are removed is due to this cause. 
The diagram Fig. 13 is taken from Dyson's paper ; it 
refers to the area R. A. 10 h to 14 h ,Dec. -30 to +30. The 
motion of each star is represented by a dot, the displace- 
ment of the dot from the origin representing a century's 
motion on the scale indicated. The blank space round the 
origin is, of course, due to the omission of all motions less 
than 20" ; we can imagine it to be filled with an extremely 



io6 STELLAR MOVEMENTS CHAP. 

dense distribution of dots. It is clear that the distribution 
shown in the diagram represents a double streaming 
approximately along the axes towards 6 h and S. No 
single stream from the origin could scatter the dots as 

o o 

they actually are. Although the general displacement is 
towards the solar antapex (i.e., towards the lower right 
corner), this is accompanied by an extreme elongation of 
the distribution in a direction almost at right angles. 
Thus the phenomenon of two streams is well shown in the 
largest, and proportionately most trustworthy, motions that 
are known, so that it requires no particular .delicacy of 
observation to detect it. 

RADIAL MOTIONS 

Doubtless the most satisfactory confirmation of this 
phenomenon found in the transverse motions of the 
stars would be an independent detection of the same 
phenomenon in the spectroscopically measured radial 
velocities. Although great progress has been made in the 
determination and publication of radial velocities, the 
stage has scarcely yet been reached when a satisfactory 
discussion of this question is possible. We shall see that 
the results at present available are quite in agreement 
with the two-stream hypothesis, and afford a valuable 
confirmation of it in a general way ; but a larger amount 
of data is required before we can see how precise is the 
agreement between the two kinds of observations. 

In the transverse motions the two streams were detected 
by considering the stars in a limited area of the sky ; there 
was no need to go outside a single area, except at a later 
stage when it was desired to show that the different parts 
of the sky were concordant. But with the radial veloci- 
ties, we can learn nothing of star-streaming from a single 
region ; that is the drawback of a one-dimensional 
projection compared with a two-dimensional. To pass 
from one area to another involves questions of stellar dis- 



VI 



THE TWO STAR STREAMS 



107 



tribution, which complicate the problem. In particular it 
is necessary to pay attention to spectral type. It is well 
known that the early type stars are more numerous near 
the galaxy than elsewhere ; as these have on the average 
smaller residual motions than the later types, there will be 
a tendency for the radial motions near the galactic plane 
to be smaller than near the poles. But evidence of star- 
streaming should be looked for in a tendency for the residual 
radial velocities to be greater near the vertices (which are 
in the galactic plane) than elsewhere. The two effects are 
opposed, and there is a danger that they will mask one 
another. 

By treating the different types separately the difficulty 
is avoided, but in that case the data become rather 
meagre. For Type A the results have been worked out by 
Campbell 9 who gives the following table : 

TABLE 9. 
A veraye Residual Velocities. Type A . 



Stitude Distance from Kapteyn's Vertices. 


0-30 
3060= 
6090 


030 


30-60 


60 -90" 


IS^ 


10-3 33 
11'2 4 6 


11 -735 km. per sec. 
7^ 
9-3^ 


Mean . . 


IS^ 


10'8 79 


Q.X 

y ioo 



The suffixes show the number of stars. 

The increased velocity in the neighbourhood of the 
vertices seems to be plainly marked, and it is in fair 
quantitative agreement with the results from the transverse 
velocities.* A division of the results according to galactic 

* There seems to have been some misunderstanding on this point. It is 
considered mathematically in Chapter VII. 



108 STELLAR MOVEMENTS CHAP. 

latitude is given, showing that the progressive increase is 
not dependent upon that. 

The radial velocities of the later types of spectrum have 
not yet been discussed. 



GENERAL CHARACTERISTICS OF THE MEMBERS OF THE 
Two STREAMS 

We now turn to the question whether there is any 
physical difference between the stars of the two streams. 
Are they of the same magnitude and spectral type on 
the average ? And are they distributed at the same 
distance from the Sun, and in the same proportions in 
all parts of the sky ? It is not possible on the two- 
drift theory definitely to assign each star to its proper 
drift ; all that can be said is that, of the stars moving in a 
specified direction, a certain proportion belong to Drift I. 
and the rest to Drift II. There are, however, directions in 
which the separation is nearly complete, and, by confining 
ourselves to these, we may pick out a sample of stars 
of which 90 per cent, or more belong to Drift L, and 
another sample equally representative of Drift II. Our 
samples are thus not absolutely pure, but they are 
sufficient for testing whether there is any physical differ- 
ence between the members of the two drifts. 

By thus separating the drifts as far as possible, the 
following table has been constructed to compare the magni- 
tudes of the stars. The stars are those of Boss's catalogue 
(stars of Type B being omitted). Approximately equal 
samples of each drift have been taken from each of ten 
regions ; the results for the five polar and five equatorial 
regions are shown separately in Table 10. 

The last two columns of the table agree closely ; the 
slight excess of very bright stars in Drift I. may be 
noticed, but it seems to be of an accidental character. 



VI 



THE TWO STAR STREAMS 



109 



TABLE 10. 

MmjnittnJ.es of Stars of the Two Streams. 





Polar B 


Equatorial Regions. 

'? g TvTT VIIL IX - XIL 


Total. 




I. II. V. \ 


XIII. & XVII. 




Magnitude. 










Drift I. 


Drift II. Drift I. 


Drift II. 


Drift I. 


Drift II. 


0-02-9 


16 


7 


4 


22 


11 


3-03-9 


17 


10 


10 


12 


27 


22 


4-04-9 


46 


52 


38 


38 


84 


90 


5-05-4 


50 


52 39 


52 


89 


104 


5-45-9 


99 


100 


78 


68 


177 


168 


6-06-4 


75 


72 


75 


79 


150 


151 


6-56-9 


50 


59 


57 


41 


107 


100 


7-0 


44 


51 


52 


49 


96 


100 


Variable 


7 


2 





2 


7 


4 


Total . 


404 


405 


355 


345 


759 


750 



10 



The investigation may be extended to somewhat 
fainter stars by taking the Groombridge proper motions. 
Samples taken from these give : 

Number of stars of magnitudes 

03-9. 4-04-9. 5-05-9. 6-06-9. 7'0 7'4. 7'5 7'!'. 8'0 8'4. S'5 8'9. Total. 

Drift I. 16 29 86 171 136 108 104 51 701 
Drift II. 3 23 81 169 125 113 132 61 707 



The excess of very bright stars is again noticed, but 
this is to some extent a repetition of the stars which 
occurred in the last table and is not fresh evidence. 
Further the Type B stars were not excluded in forming 
this table ; these form a considerable proportion of the 
bright stars, and their motions are known to be peculiar. 
At the other end of the table an excess of faint stars 
in Drift II. is shown, but it is not very decided. This 
may be spurious, being the effect of a greater accidental 
error in determining the directions of motion of the 
faintest stars, which tends to increase falsely the number 
assigned to the slower-moving drift. 



no 



STELLAR MOVEMENTS 



CHAP. 



The main conclusion from the two tables is that there 
is no important difference in the magnitudes of the 
stars constituting the two drifts. On the other hand 
there is possibly some significance in the fact that 
discussions of the faint stars, such as the Groombridge 
and Carrington stars, have given a higher proportion 
belonging to Drift II. than the discussions of brighter 
stars, such as those of Bradley and Boss. 

The same course may be pursued with regard to 
spectral types, though, in view of the known differences 
in the amount of the individual motions of late and 
early type stars, such a treatment is unsatisfactory, and 
the interpretation of the result is ambiguous. We give, 
however, results for four regions of the Groombridge 



stars : 



TABLE 11. 
Spectra of Stars of the Two Streams. 





Drift I. 


Drift II. 


Region. 


No. of 


No. of 


Percent- 


No. of No. of 


Percent- 




stars. 


stars. 


age of stars. stars. 


age of 




Type I. Type II. 


Type II. Type I. Type II. 


Type II. 


A 


61 


66 


52 


36 70 . 


66 


B 


95 


35 


27 


61 45 


42 


C 


61 


23 


27 


16 16 


50 


G 


58 


39 


40 


41 39 


49 



In each case the percentage of Type II. (later type) 
stars is higher in Drift II. than in Drift I. But some 
caution is needed in interpreting this table. It may 
be that the result is due to the elements of the star- 
stream motions differing from one type to another. 
We cannot tell whether the distribution of spectra 
differs from one drift to the other, or whether the 
drift-motions differ from one spectral type to another. 
This matter is still undecided ; but, knowing at least 



vi THE TWO STAR STREAMS in 

one remarkable relation between type and motion, we can- 
not ignore the second alternative. 

A safe way of stating the conclusion is, that stars of 
early and late types are found in both streams, but 
that there is a somewhat higher proportion of late types 
among stars moving in the direction of Drift II. than 
of Drift I. 

DISTANCES OF THE Two STREAMS 

It is most important to determine whether the two 
streams are actually intermingled in space. It might, for 
example, be suggested that one of the streams consists 
of a cluster of stars surrounding the Sun, which moves 
relatively to the background of stars constituting the other 
stream. The absence of any appreciable correlation between 
magnitude and drift renders such an explanation rather 
improbable, for it would be expected that the stars of the 
background would be fainter on the average than those of 
the nearer swarm. The question can, however be treated 
more definitely by using the magnitude of the proper 
motions to measure the distances of the two drifts. Hitherto 
we have only made use of the directions of the motions 
without reference to the amount ; we must now bring the 
latter element into consideration. 

Let G?! and d. 2 be the respective mean distances * of the 
two drifts ; if these are known, the theory (set forth in 
the next chapter) enables us to calculate the mean 
proper motion of stars moving in any direction. Take 
for instance Fig. 14, which refers to a region of the 
Groombridge catalogue ; t the curves are drawn so 
that the radius vector in any direction measures the 
mean proper motion in the corresponding direction. 
The velocities of the drifts and the numbers of stars 

* Unless otherwise specified the mean distance of a system of stars means 
the harmonic mean distance, or distance corresponding to the mean parallax. 
f This is the "restricted Region G," Monthly Notices, Vol. 67, p. 52. 



ii2 STELLAR MOVEMENTS CHAP. 

belonging to each have first been found by the usual 
method ; we can then draw the theoretical mean proper 
motion curves for any assumed values of d l and d. 2 . Two 
such curves are shown, viz., for d l = d 2 and for d l = d. 2 . 
The first curve A has a slight bi-lobed tendency, that is to 
say, there are two directions in which the radius vector is 
a maximum ; but it will be seen that the mean proper 
motion curve is not a very sensitive indicator of the presence 
of two streams. That does not matter for our present 
purpose. The upper part of the curve arises mainly from 
stars of Stream II., the lower part from Stream I. If we 
decrease the average distance of Stream I. and increase 




B C 

FIG. 14. Mean Proper Motion Curves. 

A. Theoretical Distribution. Two Drifts at the same Mean Distance. 

B. Theoretical Distribution. Second Drift at twice the Distance of First 
Drift. 

C. Observed Distribution. 

that of Stream II. the lower part of the curve will expand 
and the upper part shrink. This is what has happened in 
curve B. 

The remaining curve represents the results of observa- 
tion. We have purposely selected a region containing 
a large number of stars (767), so that the observed 
curve is fairly smooth ; but a mean proper motion is 
nearly always liable to large accidental fluctuations, and 
we must not expect a very close agreement with theory, it 
will be noticed that C is much more definitely bi-lobed 



VI 



THE TWO STAR STREAMS 



than the theoretical curves; the two streams are more 
prominent than was anticipated. This phenomenon is 
found not only in this particular example, but in all the 
divisions of the Groombridge motions. It suggests that 
our two-drift analysis has not succeeded in giving a 
complete account of the facts. (The ellipsoidal hypo- 
thesis fails equally in this respect.) The precise signifi- 
cance of the failure has not been made out ; we cannot do 
more than call attention to an outstanding difference. 

The difference in shape of the curves A and C renders 
a comparison somewhat difficult, but it will be recognised 
that the proportion of the two lobes is not very different, 
pointing to approximately equal distances for the two 
drifts. There is no such magnification of one lobe at the 
expense of the other as is illustrated in curve B. 

The values of d t and d 2 can be obtained by a rigorous 
mathematical solution by least squares. The results for 
this region (G) and six others are given in Table 12. 12 

TABLE 12. 
Mean Parallaxes of the Two Drifts. 





Limits of Region. Drift I. 


Drift II. 


Region. 










Dec. N. 


P . 1 Probable 
MI Error. 


1 

us. 


Probable 
Error. 















h 


* 


. " 


A 


7090 


024 2-96 0-07 


3-41 


0-13 


B 3870 


22 2 2-45 0-07 


2-40 


0-11 


r^ 


38-70 


2 6 2-39 0-08 


2-65 


0-12 


D 3870 


610 3-35 0-12 


3-23 


0-21 


E 3870 


1014 3-65 0-14 


478 


0-29 


F 3870 


14_18 374 0-23 


4-15 


0-31 


G 3870 


18-22 2-77 0-16 


2-55 


0-18 



The quantities \jhd l and l/hd. 2 are the mean parallaxes 
multiplied by a factor the value of which is probably about 
450. As, however, some large proper motions have been 

I 



n 4 STELLAR MOVEMENTS CHAP. 

excluded (the same proportion from each drift), the absolute 
parallaxes have no definite significance. It is the ratios 
that are of importance. 

The table shows that in each region the two drifts are 
at nearly the same mean distance. In only one case, 
Region E, is the difference at all significant, and even 
there the ratio, about 4:3, necessitates a very considerable 
intermixture of the two sets of stars. Moreover, Region 
E contains fewer stars than any other, and the results are 
in consequence uncertain. It is thus necessary to regard 
the two star-streams as thoroughly intermingled systems, 
and to throw aside any hypothesis which regards them 
as passing one behind the other in the same line of 
sight. 

The table also shows a variation in mean distance from 
region to region, which is greater than the variation from 
Drift I. to Drift II. This variation is found to follow the 
galactic latitude of the stars, and is due to the fact 
that we see a larger number of more distant stars the 
nearer we get to the plane of the Milky Way. As both 
drifts show this progressive change, these distant stars 
must belong to both drifts impartially. 

A similar conclusion is derived from the more recent 
and accurate proper motions of Boss's catalogue. In 
order to obtain a large enough number of stars we have 
thrown together Regions VIII., XII., XIII. and XVII. of 
the previous division, and considered the large region 
consisting of two antipodal areas each about 70 square. 
This region is one of high galactic latitude, so that the 
proper motions are comparatively large ; and, its centre 
being nearly 90 from both Apex and Vertex, the large 
extent of the area is less harmful than it would be in 
other parts of the sky. This region contains 1122 stars. 

In Table 13 the second column gives the number of stars 
moving in each 10 sector. The third column gives the mean 
proper motion of these stars ; in order to smooth out minor 



VI 



THE TWO STAR STREAMS 



irregularities, these means have been taken for overlapping 
30 sectors-. The fourth column gives the calculated 
mean proper motion, based on the known velocities and 



TABLE 13. 

Mean Proper Motions of a Region of Boss's Catalogue 
(Centre of Region, R.A. 0\ Dec. 0). 



Direction. 


1 No. of Stars. 


Mean Centennial P.M. 


Observed. 


Calculated. 


c 




94 




12-73 


12-61 


10 


90 


12-20 


12-61 


20 


88 


11-15 


12-54 


30 


66 


10-27 


12-32 


40 


79 


10-56 


11-45 


50 


50 


8-92 


10-51 


60 


41 


9-49 


9-35 


70 


34 


8-82 


8-99 


80 


33 


10-00 


8-91 


90 


30 


11-19 


9-13 


100 


25 


10-85 


9-35 


110 


33 


11-24 


9-57 


120 


34 


874 


9-57 


130 


36 


8-98 


9-35 


140 50 




9-09 


9-35 


150 25 




9-56 


8-99 


160 32 


8-90 


8'55 


170 13 


7-05 


8-12 


180 10 








190 6 








200 7 








210 


4 








220 


1 








230 


4 




5-98 


6-23 


240 


6 








250 


9 








260 


10 








270 


8 








280 


6 








290 


4 




6-42 


6-09 


300 


20 


7-33 


6-88 


310 


10 


7-83 


8-05 


320 


14 


9-17 


9-20 


330 


26 


10-40 


10-15 


340 


44 




11-27 11-16 


350 80 


12-46 


12-03 









I 2 



ti6 STELLAR MOVEMENTS CHAP. 

relative proportions of the drifts, and on the assumption 
that they are at the same mean distance. As the number 
of stars moving in directions between 175 and 285 is too 
small to give trustworthy separate mean proper motions, 
these have been combined to give one mean. 

The corresponding polar diagrams are given in Fig. 15. 
The agreement is very fair ; but, as in the previous case, 





Theoretical distribution Observed distribution 

FIG. 15. Mean Proper Motion Curve (Region of Boss's Catalogue). 

the two drifts appear more sharply in the observed curve 
than in the theoretical one. It is clear that our assump- 
tion of equal distances for the two drifts cannot be far 
wrong. A rigorous solution by least squares leads to the 
results : 

For Drift I. . JL = 6" '94 0"'10 per century 

ha 

Drift II i = 7"'38 0"-17 

Or, adopting the value of the unit l/h already found, 
viz., 21 km. per sec., the mean parallaxes are 

Drift I CT0156 0" '00023 

Drift II 0"-0166 0'''00038 

On the question whether the proportion of mixture 
remains the same in all parts of the sky, conflicting 
views have been held. We are not concerned with 
local irregularities, but with any general tendency of 



VI 



THE TWO STAR STREAMS 



one drift to prevail over a hemisphere or belt of the 
sky. It seems to be agreed that there is no systematic 
connection between the numbers and galactic latitude. 
Table 14 taken from the analysis of Boss's catalogue 
shows this clearly. The individual irregularities may 

TABLE 14. 
Division of the Stars between the Drifts. 



Boss's 
Region. 


No. of Stars. 


Ratio 
Drift II. : Drift I. 


Galactic 
Latitude. 


III. 


304 


0-60 


o 

1 


X. 


356 


0-66 


1 


VIT. 


448 


0-87 


9 


[I. 


354 


0-48 


12 


XVI. 


311 


0-69 


14 


XV. 


285 


071 


17 


I. 


371 


0-60 


27 


IX. 


275 


0-75 


29 


XI. 


365 


0-42 


31 


, IV. 


294 


0-87 


33 


XVII. 


245 


0-77 


37 


VIII. 


308 


0-67 


44 


VI. 


294 


0-52 


46 


- XIV. 


259 


0-68 


48 


XII. 


342 


0-64 


61 


V. 


274 


1-03 


66 


XIII. 


237 


0-59 


78 






| 



be partly real and partly the result of insufficient data 
or errors in the proper motions ; but there is no syste- 
matic progression. As already explained, these regions 
consist each of two antipodal areas, and hence 
the results do not allow us to test whether there is a 
difference between two opposite hemispheres of the sky. 
According to S. S. Hough and J. Halm 13 there is a con- 
siderable difference. From a discussion of the radial 
velocities of stars they deduced that the Drift If. stars 
were concentrated in the hemisphere towards the point R. A. 
3:24 , Dec. 12 ; the range of density was not explicitly 
determined but it was evidently considerable, the Drift 



n8 STELLAR MOVEMENTS CHAP. 

II. stars being relatively two or three times as numerous 
at one pole compared with the other. This result 
depended, at least partly, on an apparent general 
dilatation of the stellar system or excess of positive 
radial velocities compared with negative. At the present 
time the excess is more generally attributed to a 
systematic error in the radial velocities of certain types 
of stars, possibly attributable to a pressure displacement 
of the spectral lines. 'The result cannot, therefore, be 
regarded as trustworthy in itself. Analysis of the angular 
motions, however, confirmed the general conclusion. 

The same authors, 14 from an examination of the 
Bradley proper motions, found a maximum density 
of Drift II. at approximately O h R.A. and in a southern 
declination, possibly the south galactic pole ; they 
showed further that this inequality of distribution would 
fully account for certain anomalies found by Newcomb 
in his discussion of the precessional constant, viz., a 
difference in the results from the right ascensions and 
declinations, and a residual twelve-hour term 0"'50 
cos S cos 2a in the mean proper motions in right 
ascension. In his latest researches Halm has adopted 
a more complicated three-drift hypothesis ; his results 
(based now on the Boss proper motions) still indicate 
a very marked excess of Drift II. stars towards about 
22 h in general agreement with his former conclusions. 

To summarise this discussion of the distribution and 
characters of the stars of the two streams, we may 
say that on the whole the mixture is remarkably 
complete. Such differences as are found are difficult to 
interpret with certainty, being liable to be influenced 
by small systematic errors in the proper motions ; and 
in most cases it is scarcely possible as yet to discriminate 
between a difference in the proportion of the drifts ,-md 
a difference in their velocities. An excess of later 
spectral type stars in Drift II., and n relative excess of 



vi THE TWO STAR STREAMS 119 

Drift II. stars in the southern galactic hemisphere, are 
the most important differences found ; but there are 
indications that this crude interpretation of statistical 
results is not an adequate description of the complex 
distribution of motions in different parts of the stellar 
system and among different classes of stars. 

The recognition of two star-streams may be expected to 
throw some light on the discrepancies in earlier determina- 
tions of the solar motion. Of these one of the most 
remarkable is that due to H. Kobold, 15 who in fact was led 
by it to a partial recognition of the systematic motions of 
the stars. Using Bessel's method, Kobold found for the 
solar apex the position R.A. 269, Dec. -3 , differing by 
at least 35 from the position generally accepted. We 
have shown (p. 83) that Bessel's method depends very 
essentially on the individual motions of the stars being 
distributed nearly in accordance with the law of errors. 
As that is now known to be untrue, it is not surprising 
that the point found is very different from the real apex. 
It is easy to see that, but for the solar motion, Bessel's 
method would give a very sensitive determination of the 
vertex ; and it is really much better adapted for deter- 
mining the vertex than the apex. The point found by 
Kobold is indeed quite close to the vertex found on the 
two-drift or ellipsoidal hypotheses, the presence of the 
solar motion having produced a deviation of only a few 
degrees. There is another way of looking at the matter. 
In applying BesseFs method to determine the solar motion, 
only the line joining the apex and antapex is found, and 
there is nothing to indicate which end is the apex. If 
there are two drifts, we might expect that the line deter- 
mined by Kobold would be a sort of weighted mean 
between the corresponding lines of the two drifts. This is 
actually the case. But naturally Kobold's line is in the 
acute angle between the drift-axes, whereas the solar 



120 STELLAR MOVEMENTS CHAP. 

motion is the other mean lying in the obtuse angle 
between them. 

It was customary in early determinations of the solar 
apex to consider separately the proper motions between 
various limits of size. It was always found that the 
declination of the apex decreased as the magnitude of 
the proper motions increased. This is easily explained on 
the two-drift theory. Since Drift I. has much the greater 
velocity relative to the Sun, the stars belonging to it have 
on an average greater proper motions than the Drift II. 
stars. Thus the larger the motions discussed, the greater 
will be the proportion of Drift I. stars, and the nearer will 
be the resulting apex to the Drift I. apex. 

In like manner the higher declination of the solar apex 
of the later type stars may be accounted for by the 
increased proportion of Drift II. in the later types. The 
higher declination of the apex for faint stars, and the 
higher proportion of faint stars in Drift II. (both somewhat 
doubtful deductions from the observations) would also 
correspond. 

THREE-DRIFT HYPOTHESIS. 

From what has been said as to the relations of the 
two-drift and ellipsoidal hypotheses, it will be understood 
that we do not regard the analysis as giving more 
than an approximation to the actual law of stellar 
motions. Its importance is that it takes account of 
what is clearly the most striking feature of the distri- 
bution. It has been found, however, that one systematic 
deviation can be detected, of which neither of these 
hypotheses is able to take account ; we are indeed ready 
to take a step forward towards a second approximation. 
In comparing the observed distribution with that calcu- 
lated on the two-drift or ellipsoidal theories, it is found 
that there is always some excess of stars moving towards 
the solar antapex. This is shown in the diagrams of the Boss 



vr THE TWO STAR STREAMS 121 

proper motions (and, in fact, the Groombridge motions) 
by a bulge of the curve roughly in the antapex direction. 
This was attributed by the writer 16 to a small third stream, 
of much less importance than the two great drifts. Halm ir 
has shown, however, that it is better to re-analyse the 
motions on the assumption of three drifts. According to 
his representation, there is a third drift, Drift 0, which is 
practically at rest in space, and is thus intermediate 
between the two original drifts. Naturally in introducing 
the third drift into the analysis some of the stars originally 
included in Drift I. and Drift II. are regrouped with Drift 
0, and the elements of the two former become somewhat 
modified. This is especially so with Drift I., which moves 
more nearly in the same direction as Drift 0, and it is 
mainly at its expense that the new drift is formed. 

That Halm's interpretation of the phenomenon is the 
right one maybe seen by looking at the diagrams for Regions 
XIV. and XVI. (Fig. 9, ix and x). In most cases Drift I. 
and Drift overlap so much that they appear almost as one 
drift ; only a little extra bulge of the curve towards the 
antapex betrays the duality. Regions XIV. and XVI. are 
the two places where their directions become more open, 
and in these the three distinct streams of about equal 
importance are plainly manifest. Remembering that it is 
only in these two regions that we could expect Drift I. and 
Drift to be really separate, the evidence for the three- 
drift hypothesis becomes very strong. Moreover the third 
drift was postulated by Halm, not so much to explain 
these peculiarities, but because it seemed needed to 
reconcile results derived from different parts of the 
celestial sphere. 

The hypotheses of two-drifts and of three-drifts, or we 
should rather say the two successive approximations, may 
be compared thus : In Fig 16, the line CS, which 
represents the solar motion, is the same in each case. In 
the first diagram we have Stream I. with f of the stars 



122 



STELLAR MOVEMENTS 



CHA 



to the Sun being SA and SB. 




and Stream II. with r of the stars, their motions relative 

Since C is the centre of 
mass of the whole. CB : 
CA -- 2 : 3, and CM, CB 
are the motions of the 
two streams freed from 
the solar motion. In the 
second diagram we have 
regrouped the stars into 
three roughly equal streams, 
the motions of which rela- 
tive to the Sun are SA', 
SB', SO] their absolute 
motions are CA' , CB f 

Three Drifts 



Two Drifts 




FIG. 16. Comparison of Two-drift 
and Three-drift Theories. 



and zero, and here CA ' is 
approximately equal to 
CB'. Clearly also A'B' 
must be greater than AB, for in removing the slower 
moving members of A and B to form a new group at C, 
we increase the relative mean velocity of the remainder. 

The third drift has little or no motion relatively to 
the mean of the stars ; it may be considered to be 
practically at rest in space. This characteristic is a 
well-known feature of the stars of the Orion type, 
which are usually removed from the data, in making 
investigations of the two star-streams, because they are 
found not to participate in the drift motions.* It is 
natural to associate Drift with the Orion stars as 
sharing the same peculiarity of motion ; but it is not 
quite clear how close is the relation between them. 



* The Orion star^ \\viv removed from the writer's investigation of Boss's 
Preliminary General Catalogue, but not for this reason, which was in fact 
unknown at that time. Their tendency to form vague moving clusters was 
the main objection; further, as tin.- motions of those not included in the 
clusters is extremely small and, therefore, the direction of the motion is 
inaccurately determined, it was thought that the value of the results would 
In- improved by their removal (M<,nthli/ A"o//V/.s, vol. 71, p. 40). 



THE TWO STAR STREAMS 



123 



The elements of the three drifts determined by Halm 
from the Boss proper motions are : 

Apex. 

Speed. 
Drift I. . 



Drift II. . 
Drift O . 



R.A. 

90 
27CP 

90' 



Dec. 
J 
-49 
-36 



fcF-0-9 



From the analogy between the Drift and the Orion 
type stars, Halm was led to assume that their internal 
motions are smaller than those of the other drifts ; thus 

H 

Drift I. 




FIG. 17. Distribution of Drifts along the Equator (Halm). 

the speed given above is measured in terms of a different 
unit //. 

The additional number of constants to be determined 
in the three-drift analysis makes the results from 
individual regions very uncertain. Great variations in 
the relative proportions of the stars in the drifts were 
found ; these are partly accidental owing to the un- 
certainty of the analysis, but are doubtless also partly real. 
The diagram (Fig. 17) shows the distribution of the 
stars along the equator in different right ascensions, the 
radius drawn from S showing the number of star> 
belonging to the drift. The apparent relation between 



I2 4 



STELLAR MOVEMENTS 



CHAP. 



Drift and Gould's belt of bright stars is probably 
due to the fact that many of the brightest stars belong 
to the Orion type. 

Following the attitude we have adopted with regard 
to the other two hypotheses, we may be content to 
regard the three-drift theory as only an analytical 
summary of the distribution of stellar motions, without any 
hypothesis as to the physical existence of three separate 
systems. There is, however, an important property of 
Drift 0, which seems to make it something more than 
a mathematical abstraction. We have seen that all the 
stars of the Orion type appear to belong to it, and not 
to the two other drifts. It is true that in addition it 
contains stars of the other types, which are not in any 
way distinctive ; but that it should contain the whole 
of one spectral class seems to show that it corresponds 
to some real physical system, and places it on a some- 
what different footing from the two older drifts, for 
which we have as yet failed to find any definite 
characteristic apart from motion. 

SUMMARY OF DETERMINATIONS OF THE CONSTANTS OF THE Two STREAMS. 

True Vertex. 



Ref. 

No. 


Catalogue or 
Data used. 


Investigator. 


Vertex. 
R.A. Dec. 


Hypothesis. 


1 
2 
3 
4 
5 
6 
7 

8 


Auwers-Bradley . . . 

,, ... 

* 
Groom bridge .... 
.... 
Boss 
*5 

Zodiacal . ... 


Kapteyn . . . 
Rudolph . . 
Hough and Halm 
Eddington . . . 
Schwarzschild . 
Eddington . . . 
Charlier .... 

Eddington . . . 


C 

91 +13 
96+7 
90+8 
95+3 
93+6 
94 +12 
103 +19 

109 + (> 


Two-stream. 
Ellipsoidal. 
Two-drift. 
Two-drift. 
Ellipsoidal. 
Two-drift. 
Generalised 
Ellipsoidal. 
Two-drift. 


9 
10 
11 
12 


Large Proper Motions . 

Radial Velocitu * . . 
Boss . . 


Dyson .... 
Beljawsky . . . 
Hough and Halm 


88 +21 

86 +24 
88 +27 
not given 


Two-drift. 

Ellipsoid il. 
Two-drift. 
Two-stream. 


13 


Faint stars (7 m '0 - 13 m ). 


Comstock . . . 


87 +28 


Ellipsoidal. 



VI 



THE TWO STAR STREAMS 



125 



With regard to these it may be remarked that the 
method used in No. 7 gives greatest weight to the nearest 
stars, so that Nos. 7, 9, 10 refer mainly to the stars closest 
to our system. No. 10 is somewhat tentative, as the 
analysis used is not rigorously applicable to stars selected 
on account of large proper motions. No. 8 is probably 
affected by a systematic error in the data used. No. 1 1 
would probably be revised, if account were taken of the 
systematic error now believed to exist in the determina- 
tions of radial velocity of the Orion type stars. 

Apices of the Tn:o Drifts. 



Ref. 
No. 



Catalogue. 



Investigator. 



Drift I. 



Drift II. 
R.A. Dec. 



1 Auwers-Bradley . . . Kapteyn . . . 
3 ,, ... Hough and Halm 



4 Groombridge 



Eddington . . . 



85 -11 
87 -13 



90-19 



260 -48 
276 -41 
292 -58 

6 Boss Eddington . . . ! 91 -15 288 -64 

12 ,, Boss 96-8 290-54 

9 Large Proper Motions . Dyson . . . . j 93 - 7 246 - 64 

For the velocities of the two drifts relative to the sun 
the results are : 

No. 3 ratio 3 : 2 

,,4 1-7 and 0'5 

,,6 1-52 0-86 

,,9 ratioS : 2 

For the ratio of the minor axis to the major axis of the 
Schwarzschild ellipsoid, the determinations are : 

So. 2 ....... 0-56 

,,5 0-63 

,,7 0-51 

,,10 0-47 

13 . 0-62 



126 



STELLAR MOVEMENTS 



CH. VI 



No. 1 Kapteyn . . . 

,, 2 Rudolph . . . 

., 3 Hough and Halm 

,, 4 Eddington . . . 

,, 5 Schwarzschild . 

,, 6 Eddington . . . 

,, 7 Charlier .... 

., 8 Eddington . . . 

,, 9 Dyson .... 

,, 10 Beljawsky . . . 

,, 11 Hough and Halm 

,-, 12 L. Boss 

13 Comstock . . . 



AUTHORITIES. 

British Association Report, 1905, p. 257; 

Monthly Notices, 72, p. 743. 
A sir. Nach., No. 4369. 
Monthly Notices, 70, p. 568. 
Monthly Notices, 67, p. 34. 
Gb'ttingen Nachrichten, 1907, p. 614. 
Monthly Notices, 71, p. 4. 

Lund Observatory k Medd.elanden,' Serie2, No. 9. 
Monthly Notices, 68, p. 588. 
Proc. Roy. Soc. Edinburgh, 28, p. 231, and 29, 

p. 376. 

A sir. Nach., No. 4291. 
Monthly Notices, 70, p. 85. 
Unpublished, but see B. Boss, Astron. Journ., 

No. 629. 
Astron. Journ., No. 655. 



REFERENCES. CHAPTER VI. 

1. Kapteyn, Address before St. Louis Exposition Congress, 1904. 

2. Eddington, Monthly Notices, Vol. 67, p. 34 (Region F). 

3. Eddington, Monthly Notices, Vol. 71, p. 4. 

4. Boss, Astron. Journ., No. 614. 

5. Schwarzschild, Gb'ttingen Nachrichten, 1907, p. 614. 

6. Dyson, Nature, Vol. 82, p. 13. 

7. Eddington, Monthly Notices, Vol. 69, p. 571. 

8. Dyson, Proc. Roy. Soc. Edinburgh, Vol. 28, p. 231. 

9. Campbell, Lick Bulletin, No. 211. 

10. Eddington, Monthly Notices, Vol. 67, p. 58. 

11. Eddington, ibid., p. 59. 

12. Eddington, Monthly Notices, Vol. 68, p. 104. 

13. Hough and Halm, Monthly Notices, Vol. 70, p. 85. 
14 Hough and Halm, ibid., p. 568. 

15. Kobold, Astr. Nach., Nos. 3435, 3591. 

16. Eddington, Monthly Noticts, Vol. 71, p. 40. 

17. Halm, Monthly Notices, Vol. 71, p. 610. 



CHAPTER VII 

THE TWO STAR- STREAMS MATHEMATICAL THEORY 

Ix the preceding chapter we have described the principal 
results of researches on the two star- streams. We shall 
now consider the analytical methods used in the 



investigations. 



TWO-DRIFT HYPOTHESIS 



Consider a region of the sky, sufficiently small to 
be treated as plane, and consider the motions of 
the stars projected on it. Suppose that there is a 
drift of stars, i.e. a system, in which the motus peculiares 
are haphazard, but which is moving as a whole relatively 
to the Sun. We take, as the mathematical equivalent 
of haphazard, a distribution of velocities according to 
Maxwell's Law, so that the number of stars with 
individual linear motions between (u,v) and (u + du, 
v + dv) is 



In justification of this it may be pointed out that Max- 
well's is the only law T for which the frequency is the same 
for all directions, and at the same time there is no corre- 
lation between the x and y components of velocity. 
There are other laws, which make the motions random in 
direction ; but for them the expectation of the value of a 
component velocity u will differ according to the value of 



127 



128 STELLAR MOVEMENTS CHAP. 



the other component v ; for instance, with the law 
a large component v is likely to be accompanied by 
a large component u. Now we are not at the moment 
concerned with what law stellar motions are likely to 
follow ; that is a dynamical problem. We are rather 
choosing a standard of comparison with which to compare 
the actual distribution of motions, and that standard 
ought to be the simplest possible. It is by no means 
unlikely that large values of v may be correlated with 
large values of u ; but, if that should be the case, we 
ought to discover it as an explicit deviation from the 
simpler assumption of no correlation rather than conceal 
it in our initial formulae ; for it is an interesting result 
that has to be accounted for. In seeking for the unknown 
law of stellar velocities, we are at liberty to adopt any 
standard of comparison that we please, but there is clearly 
a special propriety in taking Maxwell's Law, as it is the 
nearest possible approach to an absolutely chaotic state of 
motion. 

In the frequency-law 



N is the total number of stars considered and h is 
a constant depending on the average motus peculiaris. 
It is related to the mean speed (in three dimensions) Q by 
the equation 

-W5 

Let 

F the velocity of the drift, taken to be along the axis of x ; 

r = resultant velocity of a star ; 

& = position angle, or inclination to Ox, of the resultant velocity 

(The velocities are all to be taken in linear, not angular 
measure.) 
Then 

u t + tf = v &+ V*-2 
dudv = r dr dti. 



VII 



THE TWO STAR STREAMS 



129 



Thus the number of resultant motions between position 
angles 6 and 6 4- dO is 



- 2 Vr cos Wr . 



Setting 
the number is 



Writing 



the following table gives the values of log/(T) 





TABLE 


15. 








The Function f 


. 






T. 


log/(r). 


T. 


log/(r). 


r. 


log/(r). 


T. 


log/(r). 


-1-2 


1-0411 


-0-3 


1-5363 


0-5 


0-1876 


1-3 


1-1520 


-1-1 


1-0874 


-0-2 


1-6046 


0-6 


0-2886 


1-4 


1-3003 


-1-0 


1-1355 


-o-i 


1-6763 


0-7 


0-3947 


1-5 


1-4555 


-0-9 


1-1856 0-0 


1-7514 


0-8 


0-5061 


1-6 


1-6177 


-0-8 


1-2378 0-1 1-8303 


0-9 


0-6232 


1-7 


1-7871 


-0-7 


1-2923 0-2 1-9131 


1-0 


0-7461 


1-8 


1-9637 


-0-6 


1-3493 0-3 


o-oooi 


1-1 


0-8751 


1-9 


2-1478 


-0-5 


1-4088 


0-4 


0-0916 


1-2 


1-0103 


2-0 


2-3393 


-0-4 


1-4711 















For a single drift the number of stars moving in any 
direction 6 is proportional to f (h V cos 6} ; and the 
equation to the theoretical single-drift curves discussed 
on p. 88 is 

r oc /(/iFcostf). 

The usual method of analysis is to compound two such 
curves pointing in different directions, adjusting the 



1 30 STELLAR MOVEMENTS CHAP. 

various parameters by trial and error until a satisfactory 
approximation to the observations is obtained. 

A mathematical method of determining the double-drift 
formula 



r = a l f(hV l 0080-0!) + a 2 /(/iF 2 cos 6- 2 ), 

which best represents the observations, without recourse 
to trial and error, has been given. 1 It worked quite 
satisfactorily for the Groombridge regions, where the stars 
were very numerous ; it is not, however, to be recommended. 
A mechanical method, which automatically gives some 
sort of answer, whether the distribution really corresponds 
to two drifts or not, is not so discriminating as the simpler 
synthetic process. 

ELLIPSOIDAL HYPOTHESIS. 

It has been seen that the principal fact of the 
phenomenon of star-streaming is the greater mobility of 
the stars along a certain line than in the perpendicular 
directions. K. Schwarzschild represents this mobility by 
assuming that the individual motions are distributed 
accordin to the modified Maxwellian law 



where, k being less than h, the u components of velocity 
are on the average greater than the v and w components. 
In two dimensions, let the number of stars with individual 
motions between (u, v) and (u + du, v + dv) be 



And let the components of the parallactic motion of 
the whole system be (U, V). The parallactic motion is 
not in general along the axis of greatest mobility Ox. 

As before, let r, 6 be the magnitude and direction of 
the resultant velocity of a star. Then 



+ TiV = k\r cos 6 -17)* + W(r sin 6 - F) 2 , 
dn do = r dr cW. 



vii THE TWO STAR STREAMS 131 

The number of stars moving in directions between 
and 6 + dd 



_ IThk^Q r r r e -r2(fc2 C os 2fl+^sin2)+2r(fc2D-cos 0+ft2p s in 0) - fc2[j2_ 7,272 
7T Jo 

Setting 



t _ 



J* 
ac=rVjp- 

The number becomes 

Nhk -WV 2 -^ ["> 

JO 

-v** e f r e- 

p J - 

The integral leads to the same function / as before ; and 
the number of stars moving in any direction is proportional 
to 



A little consideration shows that the polar curve 
r= - / (f ) will closely resemble a two-drift curve. 

f is a maximum near the direction of the parallactic motion 
( U, V) and a minimum in the opposite direction ; and the 
same is true for /(). This factor alone would give a 
curve not very different from a single drift curve. But 

the factor corresponds to an ellipse with its major axis 

along Ox. It distorts the approximately single-drift 
curve, pinching it along Oy and extending it along Ox, 
with the result that a bi-lobed curve is usually obtained. 

The method of determining the constants of an ellip- 
soidal distribution to suit the observations may be briefly 
noticed.* If we consider the stars moving in a direction 
6 and in the opposite direction 180 4- 6, p is the same for 
both directions, and simply changes sign. Thus the 

* I have made slight alterations in the procedure given by Schwarzschild 
in order to conserve the close correspondence with the two-drift analysis. 

K 2 



132 



STELLAR MOVEMENTS 



CHAP. 



ratio of the numbers of stars moving in these directions 
gives -J-f Y From the table of log/, we construct the 

following table (the logarithm is given as being more 
convenient for interpolation) : 

TABLE 16. 
Auxiliary Function for the Ellipsoidal Theory. 



6 


/tf>. 


6 


*k 


0f V(-). 


o-o 


o-ooo 


0-5 


0-779 


o-i 


0-154 


0-6 


0-939 


0'2 


0-309 


07 


1-102 


0-3 0-464 


0-8 


1-268 


0-4 0-620 






1 







This enables f to be found from the observations, and, 
when it is known, p is given by 



number of stars = -/() 

If now we take radii r^ = and r 2 = 



in the direc- 



tion 0, r will trace out the ellipse 



and r. 2 the straight line 



By drawing the best ellipse and straight line through 
the respective loci, & 2 , A 2 , U and F are readily found ; also 
the direction of greatest mobility, which is the major axis 
of the above ellipse, is determined. 

A very elegant direct method of arriving at the values 
of these constants has also been given by Schwarzschild - ; 
it appears, however, to be open to the same objection as 
the automatic method of determining the constants of the 
two star-streams. If these methods are used at all, it 
is very necessary to examine afterwards how closely the 



THE TWO STAR STREAMS 



133 



solutions represent the original observations. I have 
sometimes found them to be very misleading. 

The ellipsoidal hypothesis has been applied with great 
success by Schwarzschild to the analysis of the proper 
motions of the Groombridge catalogue. As an illustration 
we may take the region R.A. 14 h to 18 h , Dec. +38 to 
+ 70 already considered on the two-drift theory (p. 90). 
The comparison of the two hypotheses with the observa- 
tions is given in Table 17. In only two lines do the two 
representations differ from one another by more than a 
uni t. 

TABLE 17. 

Comparison of Ellipsoidal and Two-drift Hypotheses ivith Observations. 



T\' 


Number of Stars. 


"r^i %.y*rt 


Number of Stars. 


Direc- 
tion. 


Ob- 
served. 


Ellip- 
soidal. 


Two 

Drift. 


.Direc- 
tion. 


.Ob- 
served. 


Ellip- 
soidal. 


Two 

Drift. 


5 


4 


5 


6 


205 


21 


22 22 


15 


5 


6 


7 


215 


27 


25 


26 


25 


6 


7 


8 


225 


29 


26 


27 


35 


9 


9 


10 


235 


26 


27 


26 


45 


10 


11 


11 


245 


19 


23 


22 


55 


14 12 


12 


255 


17 


18 


18 


65 


14 13 


12 


265 


12 


14 


14 


75 


14 14 


13 


275 


11 


10 


10 


85 


13 13 


13 


285 


11 


8 





95 


12 13 


12 


295 


8 


6 6 


105 


10 12 


13 


305 


7 


5 5 


115 


11 11 


12 


315 


6 


5 4 


125 


10 10 


11 


325 


6 


4 


5 


135 


10 10 


9 


335 


5 


4 5 


145 


7 10 


9 


345 


5 


5 


6 


155 


9 11 


9 


355 4 5 6 


165 


9 


12 


11 




175 


14 


14 


12 








185 


14 16 


15 








195 


16 19 


19 
















Whilst the two hypotheses can yield closely similar 
distributions of motion as regards direction, it is conceiv- 
able that if the magnitudes of the motions were taken 
into account the resemblance might fail. But it is not 



134 



STELLAR MOVEMENTS 



CHAP. 



difficult to see that the two laws express distributions of 
linear velocities very similar in most respects, though by 
the aid of different mathematical functions, provided that 
the numbers of stars in the two drifts are practically 
equal. Schwarzschild's method is somewhat analogous to 
replacing two equal intersecting spheres by a spheroid. If 
we have two equal drifts, with velocities + V and V 
referred to the centre of mass of the whole, the frequency 
of a velocity (u, v) is proportional to 



e - 



or to 



The ellipsoidal law may be written 



The difference in the two laws is thus determined by the 
difference between cosh au and e 13 " 2 , functions which have 
a considerable general resemblance. 

As both laws give the same distribution of the v com- 
ponents, we may confine attention to the u components. 
As a typical example for comparing the laws (corresponding 
approximately with what is actually observed), we will 

k I 

take h V= 0'8, ^ = 0*58, ^ = 20 km. per sec. 

TABLE 18. 
Comparison of Two-drift and Ellipsoidal Hypotheses. 



Component 


Frequency. 


Difference 


u. 
Km. per sec. 


Two-drift 
Hypothesis. 


Ellipsoidal 
Hypothesis. 


Ellipsoidal - 
Two-drift. 





1-055 


1*160 


+ 0-105 


10 


1*099 


1-066 


-0-033 


20 


1-000 


0-828 


-0-172 


30 


0-618 


0-544 


-0-074 


40 


0-237 


0-302 


+0-065 


50 


0-056 


0-141 


+ 0-085 


60 


0-008 


0-056 


+ 0-048 


70 


o-ooi 


0-019 


+ 0-018 



VII 



THE TWO STAR STREAMS 



The two curves are shown in Fig. 18. 

Whilst there is a marked general resemblance, the 
differences are not altogether negligible. In particular 
the ellipsoidal law gives a considerably greater number 
of large velocities ; it seems probable that in this respect 
it is better fitted to the observations. The two-drift 
law gives a defect of both very small and very large 
motions, as compared with the simple error law ; this 
is sometimes expressed by saying that it has a negative 
excess. 

It may further be noticed that, although in the 
example given the frequency of the two-drift distribution 



\ 



-80 



-60 



-40 



40 



60 



80 



-20 20 

Km. per sec. 

FIG. 18. Comparison of Two-drift and Ellipsoidal Hypotheses. 
Two-driftfull curve ; Ellipsoidal dotted curve. 



makes a slight dip at the origin u = 0, for rather 
smaller values of hV this dip disappears, and the 
distribution actually agrees with the ellipsoidal distribu- 
tion in having a maximum at the origin. 3 

When the restriction that the two drifts have equal 
numbers of stars is removed, the* ellipsoidal hypothesis 
cannot approximate to the two-drift hypothesis so closely. 
There is no longer a fore-and-aft symmetry, so that the 
ellipse is an unsuitable figure to represent the frequency. 
The two-drift theory, having one additional disposable 
constant, is now able to give a considerably better 
representation of the observations. We have seen that 
the Groombridge proper motions can be represented by 



136 STELLAR MOVEMENTS CHAP. 

two drifts with approximately equal numbers of stars ; 
these can be replaced by an ellipsoidal distribution with 
practically the same precision. The Boss proper motions, 
on the other hand, require a mixture of the drifts in 
the proportion of about 3:2; the ellipsoidal hypothesis 
cannot be adapted to this skewness and accordingly fails 
to represent these observations. For this reason it has 
not been possible to analyse these more recent proper 
motions on Schwarzschild's theory.* On the other hand 
the main teaching of that theory remains, viz., that the 
dissection into two drifts may be only a mathematical 
procedure, and that it is possible to regard the distribu- 
tion of velocities as one whole. 

COMBINATION OF RESULTS FROM DIFFERENT REGIONS OF 

THE SKY. 

In the case of the two-drift theory the procedure is 
very simple. Let X^ Y^ Z l be the components of the 
velocity in space of one of the drifts, measured in the 
usual unit l/h ; v 1 and 0,, its velocity and position angle,t 
determined for a region whose centre is at (a, 8). We 
have for each region equations of condition 

t 

v t sin &i X^ sin a + Y l cos a, 

v l cos 6 l = - X l cos a sin 6 - Ysin a sin d + Z^ cos fi, 

from which X lt Y lt Z^ can be found. 

On the ellipsoidal hypothesis the projected solar motion 
is found for each region, and the results can be combined 
in the same way. But the combination of the ellipsoidal 
constants is a more complex problem. 

It will be useful to take the general ellipsoid with three 
unequal axes, although in ordinary applications a spheroid 

* C. V. L. Charlier has developed a generalisation of the ellipsoidal theory 
which permits the skewness to be taken into account, but his method 
assumes some knowledge of the distribution of the stars in space. 

f The position angle is here measured from the meridian. 



vii THE TWO STAR STREAMS 137 

only is considered. Keferred to any rectangular axes, let 
the velocity -ellipsoid be 

>iii- + l>\r + civ 2 + 2fvw + 2givu + 2huv = 1, 

so that the number of stars with individual velocities 
between (it, v, iv) and (u + du, v + dv, tv + div) is propor- 
tional to 



Take the w direction to be the line of sight. To obtain 
the distribution of the projected velocities (u, v) we must 
integrate the above expression with respect to w from 
oo to + oo . The result is 



exp - au* + 6t- 2 4- 2huv - . du dv 

The integral in this expression is a constant and equal 

,0 



Thus the projected velocities correspond to a velocity- 
ellipse 



Xow this ellipse is the right-section of the cylinder 
parallel to the i^-axis, wliich passes through the inter- 
section of the ellipsoid 

au 2 + 6r 2 + cw- + 2fcw + 2gwu + 2huv = 1, 

and the plane 



The latter is the diametral plane conjugate to the 
zr-axis, and consequently the cylinder is the enveloping 
cylinder. 

Thus the velocity-ellipse for any region is simply the 
outline of the velocity-ellipsoid viewed from an infinite 
distance in the corresponding direction. The outline 
must, of course, not be confused with the cross-section 
which is a different ellipse. 



138 STELLAR MOVEMENTS CHAP. 

Now let the velocity- ellipsoid, transformed to its princi- 
pal axes, be 



and the line of sight be in the direction (I, m, n). The 
lengths of the axes of the enveloping cylinder (i.e., of the 
velocity-ellipse) are given by 

72 /w2 2 

+ m + n = o, 
and the direction ratios of these axes are then 



It may be noted that there will be four points of the 
sky for which the velocity-ellipse will be a circle, and the 
projected motions will be haphazard. When the ellipsoid 
is a spheroid these points coalesce with the two extremities 
of the axis, in other words with the vertices. 

The general case of the ellipsoid with three unequal 
axes is of considerable interest, because it enables us to 
allow for the possibility that the motions may have a 
special relation to the plane of the Milky Way as well 
as to the axis of star-streaming. The fact that stellar 
motions have some tendency to be parallel to the Milky 
Way was pointed out by Kobold, and in recent years by 
the investigators of radial velocities. Since the stellar 
system is strongly flattened towards this plane, the result 
seemed a very natural relation between motion and 
distribution. But the star-stream investigations have 
shown that the main tendency is not towards a general 
parallelism to the Milky Way but a parallelism to a 
certain direction in it. Whether there is any residual 
relation to the Milky Way not covered by this, is a matter 
of some interest. We could test it by making an analysis 
on the basis of a velocity-ellipsoid with three unequal 
axes, and noticing whether the two smaller axes turn out 






vii THE TWO STAR STREAMS 139 

to be equal to one another. The difficulty is that, as 
already stated, the ellipsoidal hypothesis does not give a 
satisfactory representation of Boss's proper motions, which 
would naturally be used for such a test. It seems, how- 
ever, fairly certain that the deviation from a spheroid 
must be very slight. The evidence from the radial 
motions (Table 9) points in the same direction. 

Taking the velocity-ellipsoid to be a prolate spheroid 

7,V'* 4- hftv* + to 2 ) = 1 

and the velocity-ellipse for a region, with centre at an 
anular distance % from the vertex, to be 



then, since the velocity-ellipse is the apparent outline of 
the ellipsoid, we have 



_ , 

F = ~kf V 



and 

and hence 



Thus the minor axis y of the velocity-ellipse is the same 

throughout the sky ; and the last equation expresses the 
variation of the major axis. 

Also the major axis is directed along the great circle to 
the vertex. 

MEAN PROPER MOTIONS. 

These have been used in the previous chapter to 
determine the mean distances of the two drifts. It 
would have been possible to start with the mean proper 
motions in the different directions, instead of the 
simple frequency, as data, for the purpose of exhibiting 
and analysing the star-stream phenomena. But, besides 
being much less sensitive, there is the objection noticed 



1 40 STELLAR MOVEMENTS CHAP. 

in Airy's method of finding the solar motion, that 
an average motion is likely to depend mainly on a 
few specially large values, and is very much subject to 
accidental fluctuations. To use the frequency of proper 
motions leads to much smoother results ; and further, as 
we avoid giving excessive weight to the nearest stars, the 
results should be more representative of the stars as a 
whole. It is therefore better to reserve the mean proper 
motions for obtaining new information not deducible from 
the frequencies. 

We found that on the ellipsoidal hypothesis the number 
of stars moving in directions between 6 and 6 + d9 was 
proportional to 



P -f 
Hence for the mean value of x 



The numerator of this expression is equal to 






The integrated part vanishes at both limits. We thus obtain 



But = r 
Hence if 




the mean linear motion in any direction is 



VII 



THE TWO STAR STREAMS 



141 



For a simple drift *Jp reduces to h, and % to h V cos 0, 
so that the mean linear motion is 



The values of g are given in Table 19. 



TABLE 19. 
The Function y (r). 



T. 


<j(r). 


T. 


<7(r)- 


1 ' 


*. 


-1-0 


0-565 


-o-i 


0-845 


0-8 


1-315 


-0-9 


0-589 


o-o 


0-886 


0-9 


1-381 


-0-8 


0-614 


o-i 


0-930 


1-0 


1-449 


-07 


0-641 


0-2 


0-977 


1-1 


1-520 


-0-6 


0-670 


0-3 


1-027 


1-2 


1-594 


-0-5 


0-701 


0-4 


1-079 


1-3 


1-669 


-0-4 


0-734 


0-5 


1-134 


1-4 


1-747 


-0-3 


0-768 


0-6 


1-191 


1-5 


1-827 


-0-2 


0-805 


07 


1-252 


1-6 

; 


1-908 



For determining the distances of the two drifts, equations 
of condition are formed as follows : 
Let 

r be the mean proper motion in the direction 6. 
rfj, d. 2 the unknown mean distances of the stars of the two drifts 

(i.e. distances corresponding to the mean parallaxes) 
n lt 7i. 2 the numbers of stars of the two drifts moving in a direction 6. 
These have been determined by the previous analysis of the 
directions of motion. 
F,, 0, ; Vo, Oo the velocities and directions of the drift motions 



Then 





l cos 6-6^} -- + n^(A F 2 cos - 2 ) -- 



Forming these equations of condition for successive values 

of 0. we can determine -j-r- and ^ =- by a least-squares 

hd l hd 2 J 

solution. 

On the ellipsoidal hypothesis there is only the one 
unknown d, the mean distance of the stars, to deal with. 
It can be determined by the equations of condition 

mean proper motion in direction 6 = - . a(). 

d J 



142 STELLAR MOVEMENTS CHAP. 

Or the mean proper motions may be used to effect an 
independent determination of the ellipsoidal constants 
exactly as the frequency was used. The latter procedure 
is much simplified by the aid of the following theorem : 

If in the direction 6 a radius is taken, which is the 
geometric mean between the mean proper motion in 
the direction 6 and the mean proper motion in the 
opposite direction, the radius will trace out the velocity- 
ellipse (within an extremely small margin of error). 

The mean proper motions in the directions 6 and 6+ 180 
are respectively 

* riband -Lrf-0. 

N/JP *JP 

Now f can never be as great as the ratio of the solar 
motion to the minor axis of the velocity-ellipsoid. 
Actually 0'5 is about the upper limit, but to allow an 
ample margin we shall take also 1*0. From Table 19, 



for | =0-0 */KGf(-D = 0-8862 
0-5 0-8914 

1-0 0-9049 



Thus ^/g(^) g( f ) may be taken to be constant with an 
error not greater than one in fifty in the most extreme 
case. The geometric mean of the mean proper motions 

is then proportional to r , which is the radius of the 



velocity-ellipse in the corresponding direction. 

The theorem provides a short method of finding the 
velocity-ellipse in any part of the sky, provided a fairly 
large number of observed proper motions are available. 
It is subject, however, to the drawback that there are 
generally some directions in which very few stars are 
moving, so that we have to take the geometric mean of 
two quantities, one of which is badly determined and the 
other unnecessarily well determined. For the numerous 
motions of the Groombridge Catalogue the method proved 



VII 



THE TWO STAR STREAMS 



satisfactory, and the results agreed closely with those 
found from the simple frequencies of the proper motions. 4 



TABLE 20. 
Groombridge Proper Motions. 







Ratio of Axes of Velocity-Ellipse. 


Region. 


No. of Stars. 


Mean Proper 
Motions. 


Frequency of 
Proper Motions. 


A 


585 


0-59 


0-59 


B 


862 


0-56 


0-58 


C 


516 


076 


0-70 


D 


443 


0-82 


0-81 


E 


385 


0-65 


0-72 


F 


425 


0-53 


0-61 


G 


1103 


0-66 


072 



RADIAL MOTIONS TWO-DRIFT HYPOTHESIS. 

The development of the formulae necessary for the 
study of radial motions will now be considered. The radial 
differ from the transverse motions in two respects, (l) The 
transverse motions allow us to compare the motions in 
two perpendicular directions in the same region of the 
sky ; but to learn anything as to the form of the velocity 
distribution from the radial motions it is necessary to com- 
pare the results from different regions. This is evidently 
a disadvantage, for it introduces the complication of the 
differences between galactic and non-galactic stars, local 
drifts, and so on. (2) The results come out in linear 
measure, independent of the distances of the stars. 

It will be assumed throughout that the radial velocities 
have been corrected for the solar motion, and are accord- 
ingly referred to the centre of mass of the system. 

The effect of the preferential motions in the directions 
of the two vertices will be that the radial velocities will 



i 4 4 STELLAR MOVEMENTS CHAP. 

be greater on the average near the vertices than in other 
parts of the sky. This was illustrated in Table 9 (p. 107)." 

Taking first the two-drift theory, let V l and F 2 be the 
velocities of the two drifts, referred to the centroid of the 
whole system ; a and 1 a the proportion of stars in each 
drift. 

Then 

aF, = (l-a)F 2 . 

The mean radial velocity regardless of sign near the 
vertices is for Drift I. 



and the mean radial velocity at right angles to the 
vertices is 

l 

hjir' 

Thus for the two drifts the average radial velocity at the 
vertex is to the radial velocity in a region 90 from the 
vertices in the ratio 



If, using the results of the analysis of Boss's Catalogue, we 
set 

a = 0-6 l-a = (H 

7^=075 hV 2 = 1-12, 

the ratio becomes 1727 

In Table 9 the mean observed ratio was 

15'9 km. per sec. _ j. 
9*5 km. per sec. 

and, allowing for the large size of the areas considered, the 
agreement is remarkably exact. But the confirmation is 
not quite so satisfactory as it appears at first sight, because 
the stars of Table 9 are of Type A, a type showing the 
star-streaming very strongly ; and there is no doubt that 



vii THE TWO STAR STREAMS 145 

if Type A alone had been used in the transverse motions 
higher values of hV l and hV. 2 would have been obtained. 
According to H. A. Weersrna 5 the drift-velocities for Type 
A are 



the probable error being, however, nearly 10 per cent. 
These numbers lead to the ratio 2*02. 

Considering the uncertainty both of the observed ratio 
and of the drift-constants for Type A, the discordance 
between 1*68 and 2*02 is not unduly great. 

RADIAL MOTIONS ELLIPSOIDAL HYPOTHESIS. 

Generally speaking, Schwarzschild's ellipsoidal hypo- 
thesis is the most convenient for the mathematical 
discussion of the radial motions. The first problem is 
to find the distribution of the radial velocities at a 
particular point of the sky in terms of the axes of the 
velocity-ellipsoid. It must be noticed that the distribu- 
tion of the component-velocities in any direction is by 
no means the same as the distribution of whole velocities 
in that direction. 

Let the velocity-ellipsoid referred to its principal 
axes be 

U 2 V 2 1V* _ 1 
+ r: 7 "T o J- 

- b- c* 

and let the line of sight be in the direction (, m, n). 

Referring the ellipsoid to three conjugate diameters 
two of which, a' ', V, are in the plane perpendicular to 
(/, m, n) the equation can be written 

u' 2 v" 2 , w' 2 _ , 

^ + p + c , 2 - 

and the frequency of the oblique velocity components 
w' is proportional to 



If now V be the rectangular velocity component in the 

L 



146 STELLAR MOVEMENTS CHAP. 

line of sight, p the perpendicular on the tangent plane 
normal to the line of sight, 

w' ^~c' 

Thus the frequency of a component V in the direction 
/, m, n is proportional to 

< 
that is 



The fact that the divisor is the perpendicular on 
the tangent plane is analogous to the two-dimensional 
result that the velocity-ellipse is the right section of the 
tangent-cylinder. 

To determine the velocity-ellipsoid from a series of 
measures of radial velocities, suppose first that the 
observations are approximately uniformly distributed over 
the sky. By the foregoing paragraph the mean value 
of F 2 in any part of the sky is proportional to 
a 2 / 2 + 6 2 m 2 + cV, or, referred to more general axes, to a 
homogeneous expression of the second degree in Z, m, n, 
say E. 

Form now from the observed data the coefficients 

A = SF 2 Z 2 F 

B = ZFW G 

C = 



Then 

A\* + B + O> 2 



This last expression is the moment of inertia of the 

surface r' L = E* about the plane whose normal is (\, /*, v). 

The surface is not the velocity-ellipsoid, indeed it is 

not an ellipsoid at all ; it is the inverse of the reciprocal 

* The mass beini? supposed to be distributed proportionately to the solid 
angle, or, more strictly, proportionately to the numbe|: of observations of 
radial velocity. 



vii THE TWO STAR STREAMS 147 

ellipsoid. But it is evident that it will have the same 

principal planes as the velocity-ellipsoid. Thus the 

directions of the axes of the velocity-ellipsoid are those 
of the momental ellipsoid of this surface, that is of the 
quadric 

2Gv\ 



These are given by the direction -ratios 
i i i 



GH-F(A-k) ' HF-G(B-k} ' FG-H(C-k), 

where k has in succession the values of the three roots of 
the discriminating cubic 

A-k H G =0. 

H B-k F 
G F C-k 

Since the surfaces r 2 = E and r 2 = *JE have the same 
principal planes we may use | F | instead of V 2 in form- 
ing the coefficients, so that 

A = 2 | V | I 2 , F = 2 | F mn, etc. 

This is probably a preferable procedure, since squaring 
the velocities exaggerates the effect of a few exceptional 
velocities ; just as in calculating the mean error of a series 
of observations it is preferable to use the simple mean 
residual irrespective of sign rather than the mean-square 
residual.* 

When the observed radial velocities are not uniformly 
distributed over the sky the problem is more complex, but 
there is no great difficulty in working out the necessary 
formulae. 

EFFECT OF OBSERVATIONAL ERRORS. 

The accidental errors in the determination of the proper 
motions must tend to equalise the number of the stars 
moving in the different directions, and to smooth out the 

* This is contrary to the advice of most text-books ; but it can be shown 
to be true. 

L 2 



I 4 8 STELLAR MOVEMENTS CHAP. 

peculiarities of distribution caused by star-streaming. In 
consequence, the deduced velocities of the two drifts are 
likely to be too small. An approximate calculation of 
the amount of this effect may be made by taking the 
simple case when the stars are all at the same distance 
from the Sun. 

In this case the accidental errors of the proper motions 
will simply reappear (multiplied by a constant) as the 
accidental errors of the linear motions. If the true 
frequency of a component of linear motion u be 



and the frequency of an error x in it be 



the apparent frequency of linear motion u will be 

du r J^e-^-^.^-e-^dx 
J -oo VTT \ /7r 

du TOO hk _(h2+&) X 2+2hVux-h*u2 j 

= 7= / = e ax 

V7T J - 



00 V7T 



Jilf fOO / H 2 u \2 

* ^L / *-<*+*!( ~ J3?p) dx 

VTT-' -oo 



The true frequency distribution with the constant h is 
thus replaced by an apparent distribution with a constant 
Aj where 

_!_ = !+! 

V h* fc2* 

To apply this formula we may use the determinations 
of j-j which have been made for several regions. 
Thus taking Region B of Groombridge's Catalogue 

= 2" '4 per century. 
ha 



vii THE TWO STAR STREAMS 149 

The probable accidental error of a Groombriclge proper 
motion is about 0"'7 per century. 

Therefore 

0-477 0"7 ,, l 
~k~ ~~ ^T h 

1 0-61 

k 



^ I + (0-61) 2 

h? /i 2 

J_ 1>17 

^ h 

Thus for Region B the deduced velocities of the drifts 
need to be increased in the ratio . 

For the Boss proper motions the correction is not 
so important. Taking the large region already discussed 
(p. 116) 

= 7"'2 per century. 
ML 

The probable error of a Boss proper motion = 0"'55 per 
century. 

0-477 0-55 l 

~k~ W h 

1 0-160 

/; h 

_!_ 1-013 

^ h 

The correction is only about one per cent. 

The hypothesis that the stars are at the same distance 
is very far from true, and in consequence the corrections 
here determined are only rough; but the calculation is 
sufficient to show that when the motions are small and 
not very well determined the effect of the accidental errors 
may be quite appreciable. 

Of the possible systematic errors, the most important 
are those due to an error in the adopted constant of 
precession and an error in the adopted motion of the 
equinox. The former would lead to an apparent rotation 
of the stellar system about the pole of the ecliptic ; the 



1 5 o STELLAR MOVEMENTS CHAP. 

latter to a rotation about the pole of the equator. There 
is no way of determining the constant of precession 
except by a discussion of stellar motions ; the motion of 
the equinox can, however, be determined from a discussion 
of observations of the Sun, and it is a matter for consider- 
ation how much weight ought to be attached to the solar 
and stellar determinations, respectively. The recognised 
determinations of these two constants have been based on 
the principle of haphazard velocities ; on the two-stream 
theory the solution would become extremely difficult, and 
to some extent indeterminate, if the velocities and propor- 
tions of mixture of the two streams are not exactly the 
same all over the sky. The same difficulty occurs in 
defining the constant of precession as in defining the solar 
motion ; though in this case the difficulty is a practical 
and not a philosophical one.* In practice it is, within 
reasonable limits, arbitrary how much of the observed 
motions shall be attributed to the rotation of the axes of 
reference, and how much to the heavenly bodies themselves. 
The only guide is that the residual stellar motions should 
follow as simple a law as possible. But, when it is certain 
that no really simple law is possible, this is not a condition 
that can be expressed and used analytically. When once 
the hypothesis of haphazard motions is given up, the con- 
stant of precession can only be given an approximate 
value, and there is no very satisfactory way of improving 
itt 

An investigation by Hough and Halm throws important 
light on the relation between star-streaming and the pre- 
cession-constant. It is shown by them that the inequality 
of mixture of the two drifts in different parts of the sky 

* In dynamics we know precisely what we mean by absolute rotation 
though we may not be skilful enough to detect it ; absolute translation 
cannot even be denned. 

f A determination from the stars of the Orion type, which are supposed 
to be moving at random, would be of interest, but the inequality of 
distribution and the prevalence of moving clusters would make it difficult. 



VII 



THE TWO STAR STREAMS 



accounts for certain discordances in former investigations 
of the precession. But this work does not appear to lead to 
any means of determining the constant de novo. 

Having regard to the practical impossibility of obtaining 
an accurate value of the precession-constant, there is an 
important advantage in proceeding so as to avoid the 
systematic errors arising therefrom. This is done when 
we treat two antipodal areas of the sky together ; for the 
error of precession will be in opposite directions (in space) 
in the two areas, and its effect will be wholly or partially 
eliminated in the mean result. 

THE MAXWELLIAN LAW. 

The Maxwellian or error-law plays an important part in 
the analysis both of the two-drift and ellipsoidal theory. 
The radial velocity determinations now available enable us 
to test in a direct manner how far the stellar motions obey 
this particular law. 

For Type A the following table (21) compares the 
actual distribution of the radial motions (corrected for 
the solar motion) with an error-law. 7 In order to get 
rid of most of the effect of star-streaming, stars in the 
neighbourhood of the vertices, forty in number, were 
not used. 

TABLE 21 
Radial Motions of Type A. 





Number of Stars. 


Limits of Velocity. 










Error Law. 


Observed. 








km. per sec. 






0-0 4-95 


53-4 


55 


4-95 9-95 


46-2 


47 


9-9515-95 


'38-3 


30 


15-95 25-5 


27-4 


30 


25-5 40 


67 


10 


>40 


0-2 






152 



STELLAR MOVEMENTS 



CHAP. 



A similar table (22) is given for Types II. and III. 
(F5 M). The distribution of observed velocities has 
been taken from a table given by Campbell. 8 No allow- 
ance is made for the effect of star-streaming, but that 
will not have so much influence proportionately as on the 
smaller motions of Type A. 

TABLE 22. 
Radial Motitms of Types F5 M. 





Number of Stars. 


Limits of Velocity. 










Error Law. 


Observed. 


km. per sec. 






5 


135 


162 


510 


127 


131 


1015 


114 


124 


1520 


97 


102 


2025 


78 


52 


2530 


59 


39 


3035 


42 


33 


3540 


29 


17 


4050 


30 


31 


5060 


10 


11 


60-70 


2 


7 


7080 


1 


4 


>80 





10 




1 



For Type A the agreement of the observed distribu- 
tion of the velocities with the error-law is remarkably 
close. For Types F5 M the table shows that the 
correspondence is not so good. The observed distribu- 
tion has what is technically called a positive excess, that 
is to say, there are too many small and too many large 
motions compared with the number of moderate motions. 
An increase or decrease of the modulus of the error 
distribution, with which it is compared, will make the 
agreement better at one end but worse at the other 
end of the table. A distribution of this kind would be 
obtained if we mixed together error distributions having 



vri THE TWO STAR STREAMS 153 

different moduli ; and it may therefore be supposed 
that the deviations arise from the non-homogeneity of the 
material used. It is also probable that if precautions had 
been taken to avoid the effects of star-streaming, the 
excess of large motions would have been much less 
pronounced. 

"We may conclude that in a selection of stars really homo- 
geneous as regards spectral type (and perhaps luminosity 
also) the components of motion perpendicular to the line 
of star-streaming are distributed according to the error-law, 
as required by the two-drift and ellipsoidal hypotheses. 
But, in a selection of stars made under ordinary practical 
conditions, there is likely to be an excess of very large and 
very small proper motions, and a defect of moderate 
motions. 

REFERENCES. CHAPTER VII. 

1. Eddington, Monthly Notices, Vol. 68, p. 588. 

2. Schwarzschild, Gottingen Nachricliten, 1908, p. 191. 

3. Eddington, British Association Report, 1911, p. 252. 

4. Eddington, British Association Report, 1909, p. 402. 

5. Weersma, Agtrcfhysiedl Journal, Vol. 34, p. 325. 

6. Hough and Halm, Monthly Notices, Vol. 70, p. 584. 

7. Eddington, Monthly Notices, Vol. 73, p. 346. 

8. Campbell, Stellar Motions, p. 198. 

BIBLIOGRAPHY. 

For the mathematical principles of Kapteyn's original theory, the 
two-drift theory and the ellipsoidal theory, the references are : 

Kapteyn, Monthly Notices, Vol. 72, p. 743. 

Eddington, Monthly Notices, Vol. 67, p. 34. 

Schwarzschild, Gottingen Nachrichten, 1907, p. 614. 
For the direct methods, avoiding trial and error (not recommended). 

Eddington, Mwtlihj Notices, Vol. 68, p. 588. 

Schwarzschild, Gottingen Nachrichten, 1908, p. 191. 
Other papers relating to the mathematical theory are : 

Charlier, Lund Meddelanden, Series 2, No. 8 (a generalised ellipsoidal 
theory, by correlation methods). 

Hough and Halm, Monthly Nat !(*, Vol. 70, p. 85 (application to radial 
velocities). 

Oppenheim, Astr. Nach., No. 4497 (a criticism). 

v. d. Pahlen, Astr. Nach., No. 4725 (a generalised theory). 



CHAPTER VIII 

PHENOMENA ASSOCIATED WITH SPECTRAL TYPE 

IF two stars, one of Type A and the other of Type M, 
are chosen at random out of the stars in space, it may 
be confidently predicted (l) that the Type A star will be 
the more luminous of the two, and (2) that it will have a 
smaller linear velocity than the Type M star. We say 
intentionally " out of the stars in space," because, for 
example, the stars visible to the naked eye are a very 
special selection by no means representative of the true 
distribution of the stars. The odds are considerable in 
favour of both predictions being correct, though a 
failure may sometimes occur. Similar illustrations with 
other kinds of spectra might be given. In short there 
is a conspicuous correlation, on the one hand, between 
spectral type and luminosity, and, on the other, between 
spectral type and speed of motion. The former relation 
is scarcely surprising, and some correlation would be 
expected on physical grounds, although, perhaps, not so 
close as that actually found ; but the connection between 
type and speed is a most remarkable result. 

The discovery of the latter relation has come about very 
gradually. So early as 1892, W. H. Monck l pointed 
out that the stars of Type II. had larger proper motions 
on the average than those of Type I. Further research, 
especially by J. C. Kapteyn, 2 emphasised the importance 
of this discovery. It is well shown by the stars having 



154 



CH. vin SPECTRAL TYPE 155 

excessive proper motions. In a list given by Dyson 3 of 
ninety-five stars with annual proper motions of more 
than 1", there are fifty-one of which the type of spectrum 
is known ; of these, fifty are of Type II., and only one 
(Sirius) is of Type I. Again, of those with proper motions 
exceeding 0"'5, 140 belong to Type II. and four to 
Type I. It was realised that this phenomenon did not 
necessarily signify a connection between spectral type and 
the true linear speed ; and there was a general preference 
for the less startling explanation that it was due to the 
feeble luminosity and consequent nearness of stars of the 
second type. Certain investigations of the parallactic and 
cross proper motions, as well as of the radial velocities, 
appeared to confirm this view. 

The next stage w r as reached in 1903, when E. B. Frost 
and W. S. Adams 4 published their determinations of the 
radial velocities of twenty stars of the Orion type ; it was 
shown that these stars have remarkably small linear 
velocities, averaging (for one component) only seven kilo- 
metres per second. This result seems to have been regarded 
as showing that the Orion stars w^ere exceptional ; appar- 
ently it was not suspected that this was a particular case 
of a general law. 

With the introduction of the two-stream hypothesis, and 
consequent methods of investigation, fresh light was 
thrown on the subject. It was found that the " spread " 
of the motions of the Type I. stars was less wide than 
those of Type II., the former following much more closely 
the directions of the star-streams. 5 Though other inter- 
pretations were conceivable, this seemed to indicate that 
the individual motions of Type I. were smaller than those 
of Type II. Definite evidence was at length forthcoming 
in 1910 from the results of determinations of the radial 
velocities, which clearly showed that the speeds of 
the second type stars were larger on the average. But 
the radial velocity results led to a wider generalisation. 



STELLAR MOVEMENTS 



CHAP. 



J. C. Kapteyn 6 and W. W. Campbell 7 pointed out indepen- 
dently that the average linear velocity increases continually 
as we pass through the whole series from the earliest to 
the latest types, i.e., in the order B, A, F, G, K, M. The 
following table contains the results of Campbell's discussion. 



TABLE 23. 
Mean Velocities of Stars (Campbell). 



Type of Spectrum. 


Radial Velocity. &<?$). 




km. per sec. 




B 


6-52 


225 


A 


10-95 


177 


F 


14-37 


185 


G 


14-97 


128 


K 


16-8 


382 


M 


17-1 


73 


Planetary nebulae 


25-3 


12 



The velocities for F, G, and K come in the right order, 
but it would be straining the figures too far to attach 
much importance to this. The rise from B to A and 
from A to Type II. (F, G, K), is quite well marked, and 
a rise from Type II. to M is fairly indicated. The position 
of the planetary nebulae at the end is distinctly curious. 
If we have entire confidence in the law that the speed 
increases with the stage of development, it follows that a 
planetary nebula must be regarded as a final stage- 
certainly not as the origin of a star. There is some 
justice in a remark of K. T. A. Innes 8 : "The fact that 
we have seen a star change into a nebula* ought to 
outweigh every contrary speculation that stars originate 
from nebulae." It is necessary to proceed cautiously in 
such an application ; but we seem to have within our 
grasp a new method of deciding doubtful questions as to 
the order of development of the different stages in a star's 
history. 

* Referring to the phenomena of the later stages of a Nova. 



viii SPECTRAL TYPE 157 

The residual motions given in Table 23 are corrected 
for the solar motion, but not for the star-stream motions. 
They do not therefore represent what we consider to be 
the actual individual stellar motions, as distinguished 
from the systematic motions. To remove the latter 
must affect the numbers appreciably. If, following 
Schwarzschild's hypothesis, a is the mean speed at 
right angles to the star-stream direction, and c the mean 
speed towards or away from the vertex, the mean radial 
speed at a point distant 6 from the vertex is 



and the mean radial speed over the whole sphere is 



c 2 cos 2 sin 6 dB d<j>. 

This would be slightly modified by the fact that more 
stars are observed near the galactic plane than in other 
parts of the sky, but, since the axis of preferential motion 
lies in the galactic plane, the effect of this inequality is 
minimised. 

Performing the integration, the mean radial speed 
becomes 



where 



If, for example, -- =0'5G, which is probably about 
c 

true for the Type A stars, this speed is equal to l'30a. 
So that to obtain the true motus peculiaris free from the 
effects of star-streaming we should divide the result for 
Type A given in Table 23 by 1'30. For the later 
types the velocity ellipsoid is less prolate, and the divisor 
would be smaller, about 1*15 ; Type B shows no evidence 
of star-streaming and the velocity already given may 



I 5 8 



STELLAR MOVEMENTS 



CHAP, 



remain unaltered. The mean individual speeds thus 
modified would then run 

B, 6-5 ; A, 8-4 ; F, G, and K, 13*6 kilometres per second. 

There is still a steady increase of speed with advancing 
type, though the main jump is now between A and F. 

Similar results have been obtained by Lewis Boss 9 from 
a discussion of the proper motions of stars. The method, 
which had previously been applied by Kapteyn to the 
Bradley proper motions, depends on the following prin- 
ciples. Let the proper motion be resolved into two 
components, the parallactic motion v towards the solar 
antapex and the cross proper motion T at right angles to 
it. We can determine the mean parallax of the stars of 
any type from the mean parallactic motion, by the aid of 
the known speed of the solar motion. By means of this 
mean parallax, the mean value of T, regardless of sign, can 
be converted into linear measure. These linear cross- 
motions are exactly comparable with the radial motions 
that have just been discussed. Like them they are free 
from the effects of solar motion, but are not corrected for 
the star-streaming. Boss's results, which depend on the 
excellent data of his catalogue, are as follows : 

TABLE 24. 

Min Velocities of Stars (Boss). 



Type.* 


Cross Linear Motion. 


Weight 
(No. of Stars). 








B 


6-3 " i 49b'""" j 


A 


10-2 


1647 


F 


16-2 


656 


G 


18-6 


444 


K 


15-1 


1227 


M 


17-3 


222 



* In Boss's classification, B includes Oe 6 to B 5 ; A includes B 8 to A 4 ; 
F includes A 5 to F 9. 



VIII 



SPECTRAL TYPE 159 



The close agreement with the quite independent 
evidence of the radial velocities is very satisfactory. 
Boss's results depend on the assumption that the solar 
motion is the same for all types, which is open to some 
doubt. As regards the irregularity of the progression F, 
G, K, there is little doubt that his method of excluding 
stars of excessive proper motion leads to too small a value 
of the parallactic motion as compared with the cross 
motion ; and this is especially the case for Types F and G, 
which contain by far the largest proportion of great 
proper motions. The linear motions deduced by him for 
these two types should accordingly be diminished. 

The facts here brought before us direct attention to the 
very deep-lying question, How do the individual motions 
of the stars arise ? It appears that as the life-history of a 
star is traced backwards, its velocity is found to be smaller 
and smaller. In the Orion stage it is only a third of what 
it will ultimately become. Must we infer that a star is 
born without motion and gradually acquires one ? I 
believe this is the right conclusion, although there is 
more than one loophole of escape, which deserves con- 
sideration. 

J. Halm 10 has suggested that equipartition of energy 
holds in the stellar system ; according to his view the 
Orion stars move slowly, not because they are young, but 
because they are massive. If the stars were all formed 
about the same epoch, the large stars might be expected 
to take longer to pass through their stages of development 
than the smaller stars, so that at the present time the 
more massive the star the earlier would be its spectral 
type. The main direct evidence as to the masses of the 
stars is found in a discussion of the spectroscopic binaries of 
which the orbits have been investigated. In cases where 
both components are bright enough to show their spectra 
the quantity (m^ + m*) sin 3 i can be found ; here / is the 
unknown inclination of the orbit to the plane of the sky. 



160 STELLAR MOVEMENTS CHAP. 

There are available for discussion seven binaries of Type B 
and nine of Types A G. Assuming that the mean 
value of sin 3 1 will be the same for both groups, it is 
found that 

average mass of Type B binaries _ g.g 
average mass of other types 

When the spectrum of one component only can be 
observed the quantity 

' J?!_Y.in,- 



m/ 



can be found. There are seventy-three suitable orbits of 
this kind known. These give 

average mass of Type B _ g.g 
average mass of other types 

These results indicate that the B stars are considerably 
more massive than the other types ; and the ratio 
actually agrees with that demanded by the law of equi- 
partition of energy, viz., the average mass is inversely 
proportional to the square of the average velocity. But the 
main argument for equipartition has been a theoretical 
one, depending on a supposed analogy between the 
behaviour of stars and the molecules of a gas. This 
subject will be considered in Chapter XII. ; the evidence 
there given seems convincing that the analogy of the 
stellar system with a gas system does not hold good ; 
and equipartition, if it exists, cannot be explained in 
this way. 

It seems certain that the motion of a star has not 
during the period of its existence been appreciably 
disturbed by the chance passage of neighbouring stars. 
This doctrine of non-interference leads to the conception 
that each star describes a smooth orbit (not necessarily 
closed) under the central attraction of the whole stellar 
system. Such a star will wander sometimes near the 



vin SPECTRAL TYPE 161 

centre, sometimes at a remote distance, transforming 
potential into kinetic energy and vice versd. The 
nearer it is to the centre the greater will be its speed. 
Consequently in the stellar system the average speed 
may be expected to diminish from the centre outwards. 
Ihis conclusion depends on the view that most of the 
stars are continually approaching and receding from the 
centre ; if the majority were describing circular orbits 
the speed would actually increase from the centre outwards. 
But accepting it as a possible and fairly likely condition, 
it offers another explanation of the association between 
velocity and spectral type. Suppose the Orion stars 
move slowly, not because they are young, but because 
they are very distant. The order of spectral type is 
(or was until recently) believed to be the order of 
luminosity and, consequently, for stars down to a limit- 
ing magnitude, the order of mean distance. Thus it 
may be that we are using the spectral classification as a 
distance classification, and determining a relation between 
distance and speed. 

This explanation was formerly put forward in a tenta- 
tive manner by the writer, 11 but it is given here only 
that it may be disproved. To test it, the radial motions 
of the stars of Type A were taken and grouped according 
to the magnitude of the proper motion. This grouping 
is a rough division according to distance, since the larger 
proper motions usually indicate the nearer stars. 



Centennial 
Proper Motion. 


Radiaf Velocity. | ^o. of Star, 




! 






" 


km. per sec. 




>20 


10-1 


19 


1220 


8-8 29 


812 


12-4 


38 


48 


11-6 


61 


04 


11-1 


65 



162 



STELLAR MOVEMENTS 



CHAP. 



There is here no sign of a decreasing speed with 
increasing distance. It is clear that distance cannot be 
the determining factor. 12 

We are thus thrown back on the original and straight- 
forward conclusion that the phenomenon is a genuine 
correlation between speed and spectral type, independent 
of either mass or distance. 

We have up to now been discussing the relation between 
spectral type and the individual stellar motions ; it 
remains to be considered whether the systematic 
motions vary from one type to another. It was found in 
Chapter V. that the declination of the solar apex 
depended on the type of stars chosen, being more 
northerly for the later types. It is not clear whether the 
speed of the solar motion is appreciably different. The 
following results are given by Campbell, 13 but the amount 
of data is scarcely sufficient to allow of much weight being 
attached to them : 



Type. 


Solar Velocity. 


No. of Stars. 


B 
A 
F 
G 
K 
M 


km. per sec. 
20-2 
15-3 
15-8 
16-0 
21-2 
22-6 


225 
212 
185 
128 
382 
73 



According to the two-drift hypothesis the solar or 
parallactic motion is merely the mean of two partially 
opposing drift-motions, and for a fuller understanding of 
these changes, or possible changes, of the solar motion 
reference must be made to the drifts. It has been found 
by many independent researches that the star-streaming 
tendency is scarcely shown in the Type B stars, that 



vin SPECTRAL TYPE 163 

it is most strongly shown in Type A, and it becomes less 
marked in succeeding types, though still quite prominent in 
Type K. The sudden development of the star-streaming 
in its full intensity in passing from Type B to Type A is 
a curious phenomenon, but the evidence for it is over- 
whelming. 

The question has been studied quantitatively by H. A. 
Weersma 14 from the data of Boss's Catalogue. If Fis the 
velocity of one drift relative to the other, Q the mean 
individual speed of the stars, he finds, 

For Type A ZL - 2'29 0'19 



For Types K and M . = Q'98 O'll 

Again if P be the solar motion relative to the mean of the 

stars, 

For Type A .... ?l 1'08 0'08 

For Types K and M . ?* = 0'62 0'04 
Q 2 

It was assumed in the investigation that the proportion 
in which the stars are divided between the two drifts is 
the same for Type A as for K and M, viz., 3:2. It is by 
no means certain that this is correct. 

These differences between the quantities F/ft, P/Q for 
the two groups are largely accounted for by the differences 
in ft that have already been discussed ; but it would 
appear that to reconcile the results we must have also P l 
different from P. 2 or else V l different from F 2 . Having 
regard to the probable errors the evidence for this is 
rather slight. If, for example, we put fi. 2 : HI =1*8, a 
value which appears to represent the results derived from 
the radial velocities (p. 158), then 

v l -. V 2 = 1-30 

P, : P 2 = 0-97 

which makes the solar motion about equal for the two 



164 STELLAR MOVEMENTS CHAP. 

types, and gives a real diminution of the star-stream 
velocity, in passing from Type A onwards. With a some- 
what smaller ratio ft 2 : n x we should obtain the same star- 
stream velocity, but a smaller solar motion for Type A 
than Types K and M, an equally likely explanation, 
which, moreover, receives a little support from the direct 
determinations of the solar motion already quoted. 

The view favoured by Kapteyn 15 abandons the assump- 
tion that the division between the two drifts is the same 
for all types. Instead there is a continuous increase in the 
proportion of Drift II. stars as the spectral type advances, 
and at the same time a continuous change in the direction 
of the stream motions. He considers that in the course of 
time the stream -motions have slightly changed in such a 
way that the oldest stars have deviated most and the 
youngest least, but all in a higher or lower degree, from 
the original direction and velocity. 

The conspicuous relation between the luminosities of the 
stars and their spectral types has already been touched 
upon in discussing the nearest stars. Much further 
information can be gained from investigations of the 
general mass of the stars. In consequence of the fact that 
we usually consider catalogues or selections of stars 
limited by a certain apparent magnitude, the difference in 
luminosity leads to a difference in the average distance of 
the spectral classes. In saying, as we commonly do, that the 
B stars are more remote than the A stars, we do not mean 
that there is any difference in their real distribution in 
space, but only that, when we consider stars limited 
by a certain magnitude, the selection of B stars is 
dispersed through a larger volume of space than the 
selection of A stars. 

We might hope to gain information as to the average ; 
distances, and therefore as to luminosities of the spectral j 
types, by comparing their degrees of concentration to the j 



VIII 



SPECTRAL TYPE 



165 



galactic plane. The general tendency of the stars to 
crowd to the galactic plane is explained by the oblate 
shape of the stellar system, so that we see through a 
greater depth in some directions than in others. But, 
clearly, if a class of stars is confined to a small sphere 
in the centre of the stellar system, its distribution will 
not in any way be affected by the shape of the boundary. 
Thus we find that the stars with proper motions greater 
than 10" per century show no galactic concentration ; they 
are all comparatively near to us. The greater the average 
distance of the stars, or the wider the volume of space 
through which they can be seen, the more will the oblate 
shape of the system affect them. The deficiency of stars 
in the region of the galactic poles will be more and more 
marked. Thus we may expect the amount of galactic 
concentration to be a measure of the average distance of 
the class. 

From the Revised Harvard Photometry, E. C. Pickering 16 
has determined the distribution of the stars down to a 
limiting magnitude of 6*5, arranged according to 
spectral type and galactic latitude. His results are given 
in Table 25. 



TABLE 25. 
Distribution of the Stars brighter than 6 m- 5. 





Mean 










Zone. 


Galactic B. 


A. 


F. G. 


K. 


M. 




Latitude. 










I. 


o 
+ 62-:^ 8 


189 


79 61 


176 


56 


II. 


-f-41'8 28 


184 


58 69 


174 


49 


III. 


+ 21-0 69 


263 


83 70 


212 


57 


IV. 


-f 9-2 206 


323 


96 99 


266 


77 


V. 


- 7'0 161 


382 


116 84 


239 


45 


VI. 


-22-2 158 


276 


117 100 


247 


69 


VII. 


-38-2 57 


161 


94 59 


203 


59 


VIII. 


-62-3 29 


107 


77 67 


202 


45 



1 66 STELLAR MOVEMENTS CHAP. 

The eight zones are of equal area, so that the numbers 
show directly the relative density at different galactic 
latitudes. 

Pickering's division of the spectral types was as follows : 
B = O-B8; A = B9- L \3; F = A4-F2; G = F5 - G ; 
K = G5 K2; M = K5 N. The divisions are somewhat 
different from those we have previously considered. 

Taking the degree of concentration shown in these 

o o 

tables as a measure of average distance, we should arrange 
the types in the order of decreasing distance and decreasing 
luminosity, thus : 

B, A, F^FG, K, M, 

which is identical with the order of evolution usually 
accepted. 

This agrees well with the results of Chapter III. as to 
the luminosity of the stars derived from parallax investiga- 
tions. A general decrease in luminosity with advancing 
type was there noted. Further, as the sequence B, A, F, 
G, K, M is probably the order of decreasing temperature, 
it is not surprising that the luminosity should decrease in 
the same way. 

Nevertheless, this order is undoubtedly wrong. It is 
not difficult to measure the average distances of stars of the 
spectral types by less hypothetical methods. The mean 
parallactic motion in arc is proportional to the mean 
parallax, for the true linear parallactic motion is, at least 
approximately, the same for all the spectral classes. Or, 
again, by comparing the mean cross-proper motion (at 
right angles to the parallactic motion) in arc with the 
mean cross-motion in linear measure (Table 23), an 
independent determination of the mean distance is 
obtained. Five invest i-j.-u ions on these lines may be 
cited. 17 



VIII 



SPECTRAL TYPE 



167 



TABLE 26. 
Mean Distances of the Spectral Types. 



(a) L. Boss. 


(6) J 


C. Kapteyn. 


Type. 


Parallactie 
Motion. 


No. of 
Stars. 


Type. 


Mean 
Parallax. 


No. of 
Stars. 


Oe5 B5 


2-73 


490 


B 


// 
0-0068 


440 


B8 A4 


4-08 


1647 


A 


0-0098 


1088 


A5 F9 


4-99 


656 


F, G, K 


0-0224 


1036 


G 


3-12 


444 


M 


0-0111 


101 


K 


4-03 


1227 








M 


3-29 


222 








(c) W. 


W. Campbell. 


(d) H. S. Jones. 


Type. 


Mean 
Parallax. 


No. of 
Stars. 


Type. 


Mean 
Parallax. 


No. of 
Stars. 


BO-B5 


0-0061 


312 


BO-B5 


0-0031 


11 


B8, B9 


0-0129 


90 


B8 A4 


0-0058 


188 


A 


00166 


172 


A5 F9 


0-0110 


187 


F 


0-0354 


180 


GO G5 


0-0076 


141 


G 


0-0223 


118 


G6 M 


0-0056 


140 


K 


0-0146 


.346 








M 


0-0106 


71 








(e) K. Schwarzschild. 


AP Ty P l mate Colour Index. 


Parallactie 
Motion. 


No. of Stars. 




m. 






B 


-0-65 


3-5 


64 




A 


-0-35 


2-9 


332 




F 


-0-05 


8-9 


277 




G 


+ 0-25 20-8 


150 






+ 0'55 8'6 


126 




K +0-85 


7-6 


277 






+ 1-15 


4-9 


199 






+ 1-45 4-0 


184 




M +175 4-6 


71 





The parallactic motion (centennial) is 410 times the parallax. 



i68 STELLAR MOVEMENTS CHAP. 

(a) L. Boss's results, based on the proper motions of 
stars brighter than 6 m> in his Catalogue, refer to much 
the same stars as those used in Pickering's discussion of 
galactic distribution. Unfortunately Boss rejected all 
proper motions greater than 20" per century ; this has 
not only made his values systematically too small, but it 
has had a disproportionately great etfect in the case of 
Types F and G, which include the bulk of the stars with 
excessively great motions. Accordingly the values for F and 
G need to be greatly increased. 

(b) Kapteyn's results depend on less accurate proper 
motions than the preceding. To allow for differences in 
the mean magnitudes of the different types, the values of 
the mean parallax have been corrected so as to correspond 
to magnitude 5*0. 

(c) Campbell's determination is based on the cross- 
motions. It refers to stars somewhat brighter than the 
other investigations, the mean magnitude being 4 m *3. 

(d) Jones's determination depends on the parallactic 
motions of stars between Dec. 4-73 and +90, of mean 
magnitude 6 m '8. The difference of 2*5 magnitudes between 
these stars and Campbell's accounts for the smaller paral- 
laxes found. 

(e) Schwarzschild's classification is primarily by colour- 
index. The proper motions were taken from Boss's 
Catalogue. 

All the investigations agree in showing that the mean 
parallax increases steadily from Type B to a point some- 
where about F or G, and then decreases again to a small 
value for Type M. The order of distance is thus altogether 
different from the standard order B, A, F, G, K, M. In 
particular it appears that the stars of Type M are more 
distant than any other type except B. 

How then is it that the M stars show practically no 
galactic concentration, whereas the A stars are strongly 
condensed ? Our previous explanation fails, because the 



vin SPECTRAL TYPE 169 

assumption that Type M is much less remote than Type A 
is now shown to be false. It seems necessary to conclude 
that the apparent differences in galactic distribution are 
real : that the system of the A stars is very oblate, and 
the system of Type M is almost globular. 

This leads to the following theory. The stars are 
formed mainly in the galactic plane. Type B, on account 
of the low individual speeds and the short time elapsed 
since birth, remains strongly condensed in the plane. In 
succeeding stages the stars have had time to stray farther 
from the galactic plane, and their higher velocities assist 
in dispersing them from it. In the latest type, M, the 
stars have become almost uniformly scattered, and very 
little trace remains of their original plane. We shall see 
reason in Chapter XII. to modify this hypothesis slightly. 

It will be seen that, in regard to the relation of spectral 
type both to speed and to galactic concentration, we have 
been driven to adopt the straightforward interpretation of 
the phenomena that occurs most naturally to anyone who 
has not considered the subject deeply. The correlation is 
exactly what it appears to be, and subtle suggestions as to 
its being mixed up with other effects are found to fail in 
the end. Yet I think we have been right in not jumping 
to the obvious conclusion at once ; it was necessary to 
examine, and for a time prefer, the alternative explanations, 
which, though more complex in themselves, led to a simpler 
conception too simple it now appears of the stellar 
system. 

An outstanding point of great difficulty remains, which 
may most conveniently be illustrated by the stars of 
Type M. We have been led to two opposed views as 
to their luminosity. In the parallax investigations of 
Chapter III. they were found to be the faintest of all the 
types ; in the present statistical investigations they are 
found to be the most luminous, except Type B. Our 



i yo STELLAR MOVEMENTS CHAP. 

conclusion from -the parallax investigations may be judged 
to rest on rather slight though very consistent evidence ; 
but other (less trustworthy) parallaxes confirm it, and 
moreover the K stars show a similar discordance in the two 
kinds of investigation. It may be admitted at once that 
the parallax and statistical results relate to entirely 
different selections of stars ; none of the extremely feeble 
M and K stars of Tables 3 and 5 enter into the data of 
our last discussion. Both results are probably right ; but 
it is difficult to see how they are to be reconciled. 

The leading contribution to this problem is the 
hypothesis of " giant" and " dwarf" stars put forward by 
E. Hertzsprung 18 and H. N. Russell. 19 They consider each 
spectral type to have two divisions, which are not in reality 
closely related. The one class consists of intensely luminous 
stars and the other of feeble stars, with little or no 
transition between the two classes. Assuming that the 
feeble stars are much more abundant than the luminous 
stars in any volume of space, the parallax investigations 
will take hold of the dwarfs mainly, whilst the statistical 
investigations, selecting by magnitude, will be concerned 
with the giants. This will account for the different 
luminosities ; for Type M will then denote two entirely 
different classes in the two kinds of research. 

Russell has supported this hypothesis by direct evidence 
from the parallax determinations. By his kindness I am 
permitted to reproduce his diagram (Fig. 19) of the abso- 
lute luminosities of all stars for which the necessary data 
could be obtained ; parallaxes have been used to which 
we should hesitate to attach much weight ; but the 
principal features of his diagram can scarcely be doubted. 
Below each spectral type are shown dots which represent 
on a vertical scale the absolute magnitudes (magnitude 
at a distance of 10 parsecs) of individual stars of that type. 
The large circles represent mean values for bright stars of 
small proper motion and parallax. 



VIII 



SPECTRAL TYPE 



171 



The general configuration of the dots seems to corres- 
pond to two lines thus, "\ . There would appear to be two 



M 



IP 



o x 

\ 

It 



o 






Fi<;. 19. Absolute Magnitudes of Stars (Russell). 

series of stars, one very bright and of brightness almost 
independent of the spectrum, and the other diminishing 



1 72 STELLAR MOVEMENTS CHAP. 

rapidly in brightness with increasing redness. The former 
series, corresponding to the horizontal line, are the giants, 
and the latter, corresponding to the oblique line, the dwarfs. 
For Types B and A the giants and dwarfs practically 
coalesce ; but the divergence increases to a very large 
amount at Type M. It must be remarked that the 
evidence of this diagram, convincing as it looks, does not 
compel us to divide Types K and M into two distinct 
classes. The stars of which the parallax and luminosity have 
been measured are in most cases chosen for brightness or 
for nearness (large proper motion). The two groups may 
thus result from the double mode of selection, without 
implying any real division in the intrinsic luminosities. 

For example, if the absolute magnitudes M are distri- 
buted according to the frequency law 



the stars of absolute magnitude J/ and of apparent 
magnitude greater than m are those within a sphere of 
radius r given by 

lo glo r = 0'2(m-3f), 

the volume of this sphere is proportional to r 3 or to 

1QO-6 (m-H), 

and the frequency of an absolute magnitude M among 
stars limited by the magnitude m is proportional to 

e - k'*(M - Jf )2 + 1 -38 (m - M ). 

This is an error distribution with the same dispersion as 

before but ranged about the mean value M -- ^-. 

/c~ 

Thus our two methods of selecting parallax stars would 
give luminosities clustering about two separate magnitudes 

M and M . If it is supposed that }/k increases 
/c 

with advancing type, the two diverging groups will be 
explained. From Fig. 19 it appears that the two groups 



vin SPECTRAL TYPE 173 

of Type M differ by about eleven magnitudes. Setting 
0-69/7r=ll, we have l/& = 4 m '0. For Type G the 
difference is six magnitudes, and l/k = 2'9. 

Russell has shown that the error-law, assumed for the 
absolute magnitudes, is confirmed by the observations ; 
but the modulus is smaller than that calculated, viz., 
l/k=l m '6 (corresponding to a probable deviation O m '75). 
This result may be considered to refer to the mean of all 
spectral types. 

Whilst the evidence from the directly determined 
luminosities is thus scarcely conclusive, there are several 
other indications which point to a real existence of the two 
series. Perhaps the strongest argument is a theoretical 
one. According to the well-known theories of Lane and 
Ritter, as a star contracts from a highly diffused state, 
its temperature rises until a certain concentration is 
reached, after which the loss of heat by radiation is 
greater than the gain by conversion of gravitational 
energy into heat, and the star begins to cool again. 
Recent discoveries of a new supply of energy in radio- 
active processes involve some modification of these 
theories ; but probably the general result of a tempera- 
ture increasing to a maximum, and then diminishing, may 
be accepted. Now, if the spectrum of a star depends chiefly 
on its effective temperature, it may well be that the Draper 
classification groups together stars of the same temperature 
regardless of whether they are in the ascending or des- 
cending state. The former would be highly diffused 
bodies, and the latter concentrated. The surface bright- 
ness, which depends on the temperature, being the same, 
the ascending stars with large superficial area would give 
a great deal more total light than the denser descending- 
stars. If then the stars have all much the same mass 
a conclusion supported by such evidence as is available 
we shall have two groups in each type, one of low density 
and intense total luminosity and one of high density and 



STELLAR MOVEMENTS 



CHAP. 



low luminosity. These two groups will coalesce for the 
B stars, which mark the maximum temperature reached, 
and fall wider apart the lower the temperature, just as in 
the diagram. Moreover, on the ascending side the 
increasing temperature and diminishing superficial area 
will oppose one another in their effect on the luminosity, 
so that the change of brightness from type to type will be 
small. On the descending side the decreasing surface and 
decreasing surface-brightness will both lead to a rapid 
change of luminosity from type to type. 

The determinations of density of the visual and spectro- 
scopic binaries (p. 24) favour the view that some of the 
later type stars are in a very diffused condition, and that 
others are very condensed. Table 27, due to H. Shapley, 20 
contains the determinations of density of eclipsing systems. 
Unfortunately, these are mainly early type stars, but the 
tendency to divide into two groups is well illustrated even 
in F and G. 

TABLE 27. 
Stellar Densities (Shapley). 



Density 
( Water = 1). 


B. 


A. 


F. 


G. 


K. 


>i-oo 








1 




1-00 -0-50 








1 





0-50 0-20 


1 


10 


6 


1 





0-20 -0-10 


4 


12 


1 


1 





0-10 0-05 


3 


17 











0-05 -0 02 


2 


8 











0-02 0-01 





3 


1 


1 





o-oi o-ooi 


2 














o-ooi o-oooi 











2 


1 


< o-oooi 








1 


1 






It is well-known that Lockyer's classification differen- 
tiates the stars into series of ascending and descending 
temperatures, but according to Russell the giant and 
dwarf stars do not correspond to Lockyer's criteria. 



VIII 



SPECTRAL TYPE 



With regard to the possibility of distinguishing two 
series by slight differences in their spectra an interesting 
contribution has been made by Hertzsprung. In Miss 
Maury's classification certain stars are discriminated as 
having what is known as the c character, marked by the 
very sharply defined appearance of the absorption lines. 
These stars (which are rather few in number) are found to 
have much smaller proper motions than the corresponding 
stars of the same spectral type without the c characteristic. 
With one exception (v Ursae Majoris) the motions are 
almost imperceptible, generally smaller even than those of 
the early Orion type. If, in order to allow for differing 
magnitudes, the proper motions are all multiplied so as to 
represent what would be the apparent motion of the star at 
a distance at which it would appear of zero magnitude, the 
following (condensed from Her tzsp rung's table 21 ) shows the 
results for stars with and without the c characteristic : 

TABLE 28. 
Stars with the c Characteristic. 



*" MoE. 


Normal 
P.M. 


c-Star. 


Proper 
Motion. 


Normal 
P.M. 


o. 2 Can. Maj. . . 0'03 


0-20 


v Persei .... 


0-06 


n 
1-20 


67 Ophiu. ... 0-08 


0-20 


a Persei .... 


0'09 


2-22 


Rigel O'OO 


0-41 


8 Can. Maj. . . 


0-02 


3-25 


fi Sagittarii . . 0*03 


0-41 


p Cassiop. . . . 


0-07 


3-25 


2 Camelop. . 0'05 


0-52 


y Cygni .... 


o-oi 


3-25 


rj Leonis ... 0'03 


0-39 


Polaris .... 


0-11 


3-25 


a Cygni .... O'Ol 


0-44 


T) Aquilae . . . 


0-07 


0-48 


22 Androm. . . 0'09 


072 


a Aquarii . . . 


0-07 


0-48 


a Leporis ... 0'02 


0-57 


10 Camelop. . . 


0-08 


0-48 


TT Sagittarii . . 0*14 


0-57 


d Cephei . . 


0-08 


0-48 


v Ursa; Maj. . . 1'96 


0-57 


C Geminorum . 


0-02 


0-48 


Auriga? ... 0'07 


1-20 









(The proper motions are reduced to zero magnitude as standard. ) 



The second column gives the proper motion of the c- 
stcir, the third column the average proper motion of the 



176 STELLAR MOVEMENTS CHAP. 

remaining stars of the same type in Miss Maury's classifi- 
cation, in each case reduced to zero magnitude. The first 
five stars are of Type BO to B9, the next six are from A to 
F and the remainder are in sub-divisions of types F and 
G. There are no K or M stars in the table. 

It is clear that these stars with ^-characteristic must be 
very remote and therefore (with the exception of v Ursae 
Maj.) belong to the class of " giants." It looks as though 
a beginning has been made in the direct discrimination of 
the two groups by their spectra. 

It will be seen that a very strong case has been made 
out for the recognition of two divisions in the later 
spectra] types, corresponding to widely different luminosi- 
ties. Yet there is a great difficulty in accepting Russell's 
theory in its entirety ; if it is right, it causes a revolution 
in many of the results that have been generally accepted. 
In particular we shall have to revise the supposed order of 
evolution which has hitherto been assumed. Russell's 
theory gives the complete order M x K : GI F x A! B A 2 F 2 
G 2 K 2 M 2 , where the sutfix 1 refers to the giants and 2 to 
the dwarfs. It is an essential part of his theory that the 
dwarf stars of Types M and K are too faint to appear in 
the statistical investigations of proper motions and radial 
velocities ; that is to say, types M and K must be identi- 
fied with M! and K 1} so far as these investigations are con- 
cerned. Moreover, the values of the parallactic motion show 
that the dwarfs of Types F and G play a predominant 
part in such researches, for the average distance and 
intrinsic brightness of these types is much less than for 
Types B and A. Therefore, somewhere about Type G we 
cross over, as it were, from the ascending branch to the 
descending branch. Having regard to this, the order of 
evolution becomes M K B A F G ; an order which applies 
to all researches' depending on stars selected for magnitude. 
From the astrophysical point of view the apparent breach 
of continuity between K and B does not matter ; it is not 



VIII 



SPECTRAL TYPE 177 



suggested that stars really pass from K to B at a jump, 
but that the intermediate types are outnumbered in the 
catalogues by stars indistinguishable from them in spectra 
but in a later stage of evolution. 

The new order M K B A F G upsets altogether the 
regular progression of speed with type, and of galactic 
concentration with type. We should have to suppose that 
a star is born with a large velocity, that the speed 
decreases almost to rest and then increases again. Even 
if we do not insist on the predominance of dwarfs in F 
and G (though by abandoning it we lose one of the advan- 
tages of Russell's theory), and are content with simply 
reversing the usual order of evolution, the difficulties are 
great. We should have to amend our former hypothesis, 
and suppose that the stars originate with large velocities in a 
nearly spherical distribution, and afterwards become con- 
centrated to the galactic plane, losing their velocities as 
they do so. The explanation is not improved by being 
turned upside down. And, further, it was shown in Chapter 
III. that the feebly luminous stars the dwarfs of Types K 
and M have extremely large velocities, so that in still later 
stages the speeds must increase again, even beyond their 
original amounts. The fact that both dwarfs and giants of 
Types K and M have larger speeds than the other types 
seems to imply a close connection between them, and it 
is very unsatisfactory to place them at opposite ends 
of the evolutionary scheme.* 

There is another piece of evidence which lends strong 
support to the generally accepted order of evolution. 
Table 29 shows the periods of the spectroscopic binaries 
arranged according to type ; it is due to Campbell." 

* For Russell's reply to these criticisms, see The Observatory, April, 1914, 
p. 165. 



i 7 8 



STELLAR MOVEMENTS 



CHAP. 



TABLE 29. 
Periods of Spectroscopic Binaries (Campbell). 





Period. 




Type. 




Total 




Short. 


O d 5* 


5 d 10" 


10 d -365 d 


> 1 year 


Long. 




O and B 


8 


15 


10 


14 


1 





48 


A 


4 


10 


1 


12 


2 





29 


F 





6 


2 


4 


3 


1 


16 


G 











1 


6 


3 


10 


K 











2 


3 


9 14 


M 














1 


1 2 



The columns headed " short " and " long " contain stars 
the periods of which have not been determined. 

The increase in period with advancing type is very 
striking. It is also significant that so large a proportion 
of the spectroscopic binaries are of early type, indicating 
that the later types generally move too slowly to be de- 
tected with the spectroscope. A rough classification by 
E. G. Aitken of the more rapidly moving visual double 
stars gave the following proportions : 



Types O and B 
A ,, F 
G K 

M , N 



4 stars. 
131 

28 
1 



Thus components of Type B are rarely sufficiently far 
apart to be seen separated ; and it would appear that in 
Types M and N the separation is so great that they are 
almost unrepresented on Aitken's list. If we believe that 
double stars originate by fission, and that the components 
separate farther and farther under the influence of tidal and 
other forces,* as time advances, we cannot but regard 
this result as a thorough vindication of the standard order 
of the types. Moreover, in the case of the spectroscopic 

* H. N. Russell has directed my attention to the fact that tidal forces are 
competent to produce only a limited amount of separation. 



vin SPECTRAL TYPE 179 

binaries at least, we are dealing with stars selected for 
brightness exactly as in the statistical investigations of 
stellar motions just the selection for which the order is 
challenged by Russell's hypothesis. 

To sum up the present position, there is direct evidence 
that in the later types the stars are of two grades of 
luminosity, and that they are of two grades of density. 
The former division might perhaps be due to two different 
principles of selection of the stars being employed, though 
that would leave unexplained why on the one principle 
the late type stars are the faintest, and on the other 
principle the brightest of the classes. If we associate the 
divisions of luminosity with the divisions of density, as 
the Lane-Ritter theory suggests, this upsets the usually 
accepted order of evolution, so far as statistical investi- 
gations are concerned, an order which has been indepen- 
dently confirmed by studies of stellar velocities, galactic 
distribution, and the periods of binary stars.* 

We shall conclude this chapter with some general 
remarks on the Orion type and fourth type stars, both of 
which present some interesting features. 

The position of the Orion or Type B stars is very 
remarkable. Neither from their proper motions nor from 
their radial velocities do they show any tendency to 
partake of the motions of the two star-streams. If we had 
to class them with one of the drifts by their motions, we 
should naturally assign them to Drift I., but that is only 
because the Drift I. motion approximates more nearly to 
the parallactic motion than Drift II. does. Actually, such 
systematic motion as there is seems to be purely paral- 
lactic and due to the motion of the Sun in space. We 
now know that this peculiarity of being at rest in space, 
except for small individual motions, is shared by other 

* Elsewhere in this book the point of view is always that of the older 
theory not Russell's theory unless expressly stated. 

N 2 



1 8o STELLAR MOVEMENTS CHAP. 

stars not belonging to this type. As we have seen, the 
dissection of the stellar motions into two streams leaves 
over a certain excess of stars (including both A and K 
types) moving towards the solar antapex, and presumably 
therefore at rest when the parallactic motion is removed. 

A notable feature of the Type B stars is their tendency 
to aggregate into moving clusters. " Moving Clusters " is 
perhaps rather a misnomer, for the motion is usually very 
small ; but they are groups apparently analogous to the 
Hyades cluster. The great Scorpius-Centaurus cluster, the 
constellation Orion, the Pleiades and the Perseus cluster 
between them account for a considerable proportion of the 
known stars of this type. The distance of these groups 
appears to be from about seventy to one hundred parsecs, 
except perhaps in Orion, which may be more distant. The 
remaining stars of the type have generally considerably 
smaller proper motions than these cluster stars, and are 
judged to be more distant still. This was well shown in 
Fig. 4 in which the non- cluster stars, forming the group 
of crosses near the origin, have barely appreciable proper 
motions. Lewis Boss 23 finds from his discussion of proper 
motions that a space round the Sun having a radius of 
seventy parsecs (corresponding to a parallax //g 015) is 
" almost wholly devoid of these stars." Such a space 
would, according to the conclusions of Chapter III., contain 
at least 70,000 stars of other types. There seems no 
reason to believe that the part of space around our Sun 
is unusually bare of Type B stars ; it is rather to be 
supposed that their general distribution is extremely rare, 
but that owing to brightness they are visible at say ten 
times the distance of an ordinary star and therefore 
through a space a thousand times as great. Their distri- 
bution, though rare on the average, is irregular, and in the 
moving clusters there must be many comparatively crowded 
together. 

The assumption that the proper motion gives a measure 



VIII 



SPECTRAL TYPE 



181 



of the distance of these stars is more than usually justi- 
fiable in this case. Owing to their small individual 
motions, and to the absence of star-streaming, the whole 
motion cannot generally differ much from the parallactic 
motion. Omitting the divisions B8 and B9 which are 
probably more closely allied with Type A, there is only one 
known case of a large proper motion, that of the star a 
Gruis with a motion of 20"'2. Its parallax (determined 
by Gill with great accuracy) is only 0"'024, so that its 
linear velocity must be remarkably great for its class. No 
other stars from BO to B7 have centennial motions so great 
as 10". Of late Type B stars (B8 and B9) the motions of 
Regulus, 25", and of 13 Tauri. 18" per century, are unusually 
large. 

The fourth type stars (Type N) are for the most part 
too faint to come into the general discussions of distribution 
and motions. In Pickering's table of the galactic distribu- 
tion of the types, the few that were included were classed 
with the M stars. Happily there are too few of them to 
affect the figures appreciably, for they present a very 
strong contrast to Type M in their distribution. It has 
been shown by T. E. Espin 24 and J. A. Parkhurst 25 that 
they are strongly concentrated to the galactic plane, as the 
following table shows : 

TABLE 30. 
Galactic Distribution of Type N. 





Number of N -Stars. 


Relative Density. Density 


Galactic 






of Durch- 


Latitude. 








musterung 




Espin. 


Parkhurst. 


Espin. 


Parkhurst. 


Stars. 


05 1 
5-10 / 


123 


( 92 1 
1 46 1 


11-4 


( 18-3 
\ 9-2 


27 
2-6 


1020 43 


58 


4-0 


6-0 2-1 


20-30 27 17 


3-0 


1-9 1-5 


>30 31 29 


1-0 


i-o i-o 



1 82 STELLAR MOVEMENTS CHAP. 

The concentration is even a little stronger than for the 
Orion stars, but allowance should be made for the fact 
that we are using a much fainter limit of magnitude 
for these than for Pickering's table. The fainter the 
magnitude and the greater the distance, the stronger 
should be the apparent concentration to the galactic plane, 
as is indeed borne out by observation. Making allowance 
for this, Type N may probably be placed between B and A 
in the order of galactic condensation. 

From 120 stars of this type of average magnitude 8 m< 2, 
J. C. Kapteyn 26 has determined the parallactic motion ; this 
is found to be 0"'30 per century with a probable error of 
practically the same amount. The corresponding parallax 
would be 0"'0007itO"-0007. For the Orion stars the paral- 
lax found by the same method was 0"-00680"*0004 for 
magnitude 5'0 (agreeing with other determinations already 
quoted). These N stars are thus many times more remote 
than the Orion stars. Their luminosity, however, may be 
slightly less, the difference 3 m< 2 in apparent magnitude 
counterbalancing the greater distance. 

Hale, Ellerman, and Parkhurst 27 have pointed out that 
the fourth type stars possibly have certain features in 
common with the Wolf-Rayet type. But they saw no 
reason to believe that any important organic relationship 
<>! meets the two types. 

REFERENCES. CHAPTER VIII. 

1. Monck, Astronomy and A*ti-<>p/ii/>c8, Vols. 11 and 12. 

2. Kapteyn, Astr. Nach., No. 3487. 

3. Dyson, Proc. Roy. Soc. Edinburgh, Vol. 29, p. 378. 

4. Frost and Adams, Yerkes Decennial Publications, Vol. 2, p. 143. 

5. Eddington, Nature, Vol. 76, p. 250 ; Dyson, loc. cit., pp. 389, 390. 

6. Kapteyn, Astrophysical Journal, Vol. 31, p. 258. 

7. Campbell, Lick Bulletin, No. 196. 

8. Innes, The O/^-,-,-,//,,,-,/, v..l. :',>, p. 270. 

9. Boss, Astron. Jcmrn., Nos. 623-4, p. 198. 

10. Halm, Monthly Notice*, Vol. 71. p. ;:;i. 

11. Eddington, /*///. Auoc. /.v,,,,,-/, 1911, 'p. 259. 

12. See also Kapteyn, Proc. Amnh'nlum JW., 1911, pp. 528, 911. 



vin SPECTRAL TYPE 183 

13. Campbell, Lick Bulletin, No. 211. 

14. Weersma, Astrophysical Journal, Vol. 34, p. 32 -V 

15. Kapteyn. Pr<><-. Amsterdam Acad., 1911, p. .VJ4. 

16. Pickering, Harvard A;mals, Vol. 64, p. 144. 

17. L. Boss, A.ttrnn. Journ., Nos. 623-4 ; Kapteyn, Astrophysical Journal, 
Vol. 32, p. 95; Campbell, Lick Bulletin, No. 190, p. 132; Jones, Monthly 
Notices, Vol. 74, p. 168 ; Schwarzschild, Gottingen Aktwometrie Teil B., 
p. 37. 

18. Hertzsprung, Zeit. fur. Wiss. Phot., Vol. 3, p. 429 ; Vol. 5, p. 86 ; 
Axtr. Nach., No. 4296. 

19. Russell, The Observatory, Vol. 36, p. 324 ; Vol. 37, p. 165. 

20. Shapley, Astrophysical Journal, Vol. 38, p. 158. 

21. Hertzsprung, Zeit. fur. Wins. Phot., Vol. 5, p. 86. 

22. Campbell, Stellar Motions, p. 260. 

23. Boss, Astron. Journ. t Nos. 623-4. 

24. Espin, Astrophysical Journal, Vol. 10, p. 169. 

25. Parkhurst, Yerkes Decennial Publications, Vol. 2, p. 127. 

26. Kapteyn, Astrophysical Journal, Vol. 32, p. 91. 

27. Hale, Ellerman, and Parkhurst, Yerkes Decennial Publications, 
Vol. 2, p. 253. 



CHAPTER IX 

COUNTS OF STARS 

IN the investigations described in the four preceding 
chapters we have generally been confined to stars brighter 
than the seventh magnitude. Occasional excursions have 
been made beyond that limit, and stars down to the ninth 
or tenth magnitudes have made some contribution to our 
knowledge ; further than this we have been unable to go. 
Beyond the tenth magnitude there is an ever-increasing 
multitude of stars, which hold their secret securely. We 
know nothing of their parallaxes, nothing of their spectra, 
nothing of their motions. There is only one thing we can 
do count them. Carefully compiled statistics of the 
number of stars down to definite limits of faintness can 
still yield information which is of value for our purpose. 

The fundamental theorem relating to these statistics is 
as follows : 

In a stellar system of unlimited extent in which the 
stars are scattered uniformly, the ratio of the number of 
stars of any magnitude to the number of stars one 
magnitude brighter is 3 '98. 

The ratio alluded to is usually called the star-ratio. If 
in any direction it is found that the star-ratio falls below 
the theoretical value 3*98, this shows that we have 
penetrated so far as to detect a thinning out in the density 
of distribution of the stars. It is assumed that the 
absorption of light in space is negligible. 



184 



CH. ix COUNTS OF STARS 185 

The number 3 '98 is equivalent to (2*512)* and the 
formula may be written : 

star -ratio for one magnitude = (light-ratio for one magnitude)?. 

In this form the theorem becomes fairly evident. A light 
ratio of 2*512 involves a distance-ratio of (2*512)^, and a 
volume-ratio (2*512)*. That is, for every small volume of 
space S at a distance D, there will be a corresponding 
volume (2'512)*ata distance (2'512)iD, such that the 
distribution of apparent magnitudes of the stars in the two 
volumes will correspond except for a difference of one 
magnitude due to the distance factor. But there will be 
(2 '5 12)* times as many stars in the second volume as in 
the first. Hence, a drop of one magnitude multiplies the 
number of stars by the factor 3 '9 8 through the whole 
range. The fact that the stars are of varying degrees of 
intrinsic brightness is taken into account in this 
argument. 

The thinning-out of the stars at great distances from the 
Sun manifests itself in a gradually-decreasing value of the 
star-ratio for successively fainter magnitudes. It is of 
great importance to have accurate knowledge of the rate 
at which the star-ratio diminishes, and particularly of the 
way in which it is related to galactic latitude. It may be 
hoped that the information may lead to a more precise 
knowledge of the flattening of the stellar system towards 
the plane of the Milky Way. 

The value of any compilation of star-counts will depend 
mainly on the accuracy with which the magnitudes, to 
which they refer, have been determined. In modern 
researches, the standardising of the counts by a special 
photometric investigation is a sine qua non. But it is 
only recently that this refinement has come within 
practical possibilities ; and much statistical matter that 
has been used up to now depends on the ingenious adapta- 
tion and correction of data, which were initially rather un- 



1 86 STELLAR MOVEMENTS CHAP. 

suitable. It must be admitted that these early researches 
accomplished their end in the main ; and that they have not 
only prepared the way for more satisfactory determinations, 
but also have taught us much with regard to stellar distri- 
bution that has a lasting value. But having now available 
sufficient statistics based on sound magnitude-standards, 
we shall not need to recur to the pioneer discussions, 
except where discrepancies of particular interest arise. 

An investigation of the number of stars of each magni- 
tude by S. Chapman and P. J. Melotte, 1 published in 1914, 
contains by far the most comprehensive treatment of this 
problem, and we shall attach the greatest weight to it. 
The magnitudes are photographic magnitudes based on the 
Harvard Standard North Polar Sequence. The general 
accuracy of the Harvard magnitude-scale has been 
confirmed by investigations made at Mount Wilson and 
Greenwich ; and for our present purposes it is believed to 
be accurate enough ; it is possible, however, that the 
corrections may not be altogether negligible in future 
more elaborate discussions. The statistics given by 
Chapman and Melotte extend from magnitude 2 m '0 to 
17 ra *0. This huge range (representing a light-ratio of 
1,000,000 to 1) is filled in almost continuously by data 
derived from five separate investigations. As each 
investigation is particularly strong near the middle 
point of its range, there are five well-determined points. 
These alone should suffice to give a correct idea of the 
course of the star-numbers throughout the fifteen magni- 
tude intervals, even without the weaker results, which 
bridge the gaps. 

The five sources of data are as follows : 

(1) Magnitude 12 to 17*5. Counts on the Franklin- 
Adams chart of the sky. These contain the results from 
750 areas scattered over the northern hemisphere, each 
containing from 60 to 90 stars in all. This represents 
only a portion of the counting of the Franklin-Adams 



IX 



COUNTS OF STARS 187 



chart carried out at Greenwich; but for the other areas 
the comparison with the standard sequence has not yet 
been effected, and, accordingly, the results are not used. 

(2) Magnitude 9 to 12*5. Counts of stars in the 
Greenwich Astrographic Catalogue (Dec. +64 to +90). 
For 19f) plates the formulae for reducing the measured 
diameter, published in the catalogue, to magnitude had 
been determined by rigorous comparison with the standard 
sequence, and these results were used for the present 
investigation. 

(3) Magnitude 6*5 to 9. Counts of stars in the Green- 
wich Catalogue of Photographic Magnitudes of Stars 
brighter than 9 m '0 between Declination + 75 and the Pole. 
These magnitude-determinations were made from plates 
specially taken for the purpose with a portrait-lens, the 
Astrographic plates being unsuitable for stars of this 
brightness. 

(4) Magnitude 5 to 7 '5. Counts of stars in Schwarz- 
schild's Gottingen Actinometry for Dec. to + 20. A 
small correction (O m * 13) was required to reduce from the 
Gottingen to the Harvard scale of magnitudes. 

(5) Magnitude 2*0 to 4*5. Counts of stars in the 
Harvard catalogue of photographic magnitudes of bright 
stars (Harvard Annals, Vol. 71, Pt. I.). This is the least 
satisfactory part of the data, for the magnitudes were not 
determined photographically, but were found from the 
visual magnitudes by applying the colour index corre- 
sponding to the known spectral type of each star. The 
magnitudes were published before the standard sequence 
appeared, and it is not clear how far they conform to 
that scale. 

As the principal feature in the apparent distribution of 
the stars is the variation with galactic latitude, the data 
have been arranged in eight galactic belts. The first seven 
belts are from 10, 1020, .... 60 70, and 
belt VIII is from 70 90 galactic latitude (North or 



i88 



STELLAR MOVEMENTS 



CHAP. 



South). The sources of data (l). (4), (5) cover the whole 
range, but (2) and (3) are confined to the belts I V 

Magnitude 

5 6 7 8 9 10 11 12 13 14 15 16 17 
I I I I I I I I I III 



- log B 



m 




I I I I I 
12 13 14 15 16 



I I I I I I 
5 6 7 8 9 10 11 

Magnitude 

Fio. 20. Number of stars brighter than each magnitude for eight zones. 
(Chapman and Melotte). 

and II V respectively ; thus the information is not so 
complete for the three highest zones as for the remainder. 



IX 



COUNTS OF STARS 



189 



If B m is the number of stars per square degree brighter 
than the magnitude w, the counts are most conveniently 
exhibited by plotting log B m against ra. This is done for 
each of the eight belts in Fig. 20, and smooth curves have 
been drawn to represent the results. In order to prevent 
overlapping of the eight curves, they have been displaced 
successively through an amount O'o in the vertical direc- 
tion. The lowest curve corresponds to belt I. The 
alternation of dots and crosses serves to differentiate the 
four sources of data. The data (5) are not shown. 

It is the central part of the data from each source that 
is best determined, and the outer parts may be expected 
to run off the curve a little. The general agreement of 
the separate sets of data is very satisfactory. 

The values of log B m for each magnitude, as read from 
the curves, are given in Table 31. 

TABLE 31. 
Log B m for each Magnitude (Chapman and Melotte). 



Zone. 


I. 


II. 


III. 


IV. 


V. VI. VII. VIII. 


Whole 
Sky. 


G'ilElCtlC 














Latitude. 


10 


10' 20" 


20' 30 


30 40' 


40 50 50 60 60 70 70* 90 


90 


Magnitude 














ra. 














5-0 


2-435 


2-360 


2-170 


5-055 


2-040 2-030 2-105 2'120 


2-223 


6-0 


1-010 


2-950 


2-800 


2-700 


2-655 2-610 2-660 2'680 


2-819 


7-0 


1-555 


1-500 


] -385 


1-295 


1-215 1-170 1-180 1-200 


1-377 


8-0 


0-065 


0-015 


1-930 


1-830 


1-730 1-685 1 670 1*670 


1-895 


9-0 


0-545 


0-490 


0-420 


0-320 


0-200 0-165 0-125 0'115 


0374 


10-0 


0990 


0-935 


0-880 


0-770 


0-640 0-605 0-550 0'520 


0-819 


11-0 


1-405 


1 -345 


1-300 


1-180 


1-030 1-010 0-940 0-900 


1-229 


12-0 


1-790 


1 7'2.-> 


1-680 


1 -.54.5 


1-385 1-385 1-305 1-2.55 


1-605 


13-0 


2-150 


2-075 


2-020 


1-880 


1-715 1-730 1-645 1*585 


1-951 


14-0 


2-485 


2-405 


2-340 


2-185 


2-020 2-045 1965 1-890 


2-268 


15-0 


2-800 


2-715 


2-630 


2-465 


2-300 2-33.5 2-265 2190 


2-575 


16-0 


3-095 


3005 


2-900 


2-720 


2-565 2600 2 -.540 2 "470 


2-855 


17-0 


3-380 


3-285 


3-15.5 


2-965 


2-815 2-850 2-810 2'74.5 


3-12.5 



It is of interest to find the total number of stars in 
the sky, so far as can be deduced from the samples dis- 
cussed. The results are as follows : 



190 



STELLAR MOVEMENTS 



CHAP 



TABLE 32. 

Number of Stars in the Sky brighter than a given Magnitude 
(Chapman and Melotte). 



Limiting Number of 


Limiting 


Number of 


Magnitude. Stars. 


Magnitude. 


Stars. 


m. 


m. 




5-0 


689 12-0 1,659,000 


6-0 


2,715 13'0 3,682,000 


7-0 


9,810 14-0 7,646,000 


8-0 


32,360 


15-0 15,470,000 


9-0 


97,400 


16'0 29,510,000 


10-0 


271,800 


17-0 54,900,000 


11-0 


698,000 







The curves in Fig. 20 are approximately parabolic arcs, 
and the results can be expressed with satisfactory accuracy 
by empirical formulae of the type, 



(1) 



But it is more convenient to use m 11 instead of m, since 
the zero of magnitude is outside the range we are con- 
sidering. The formulae for the eight zones are as 
follows : 

TABLE 33. 
Number of Stars brighter than a gicen Magnitude. 



Zone I. log 10 B w = 1'404 


+ 0-409 (m-11) -0-0139 (m-11) 2 


II. . . 


= 1-345 


+ 0-407 


-0-0147 






III. . . 


= 1-300 


+ 0-411 


-0-0193 






IV. . . 


= 1-177 


+ 0-403 


-0-0186 






V. . . 


= 1-029 


+ 0-391 


-0-0168 






VI. . . 


= 1-008 


+ 0-399 


-0-0160 






VII. . . 


= 0-941 


+ 0-389 


-0-0135 






VIII. 


= 0-901 


+ 0-380 


-0-0130 





The formulae make it clear that the main part of the 
variation with galactic latitude consists in a change of the 
constant term. The coefficient of (m 11) is nearly 
stationary, and that of (m II) 2 shows no systematic 
progression with latitude. The ratio of the star-density 



ix COUNTS OF STARS 191 

near the galactic pole to that near the galactic plane 
is practically the same for all magnitudes. Or, taking 
another point of view, the rate of increase in the number 
of stars with advancing magnitude follows the same law 
for all latitudes. 

This conclusion is, of course, only approximate. We see 
from Table 31 that the ratio of the star-density in Zone 1 
to that in Zone VIII is 

For magnitude 6 m '0, 2'1 : 1 
17 m '0, 4-3 : 1 

from which the conclusion may be drawn that the rate 
of increase in the number of stars is appreciably greater 
near the galactic plane than away from it. But these 
differences are very slight compared with those found in 
some former investigations, which have hitherto found 
wide acceptance. In particular the celebrated star-gauges 
of the Herschels, which have dominated our views as to the 
distribution of faint stars for nearly a century, gave widely 
different results. Prior to Chapman and Melotte's work 
the most extensive discussion of magnitude statistics was 
that of J. C. Kapteyn 2 (1908), which gave a very different 
idea of the effect of galactic latitude on the star-density. 
From Kapteyn's table it appears that the ratio of the star- 
density in Zone I to that in Zone VIII should be 

For magnitude 6 m '0, 2'2 : 1 
17 m '0, 45 : 1. 

It is difficult to explain the huge difference between 
Kapteyn's result for the distribution of the faint stars, and 
that of Chapman and Melotte. The former research was 
somewhat provisional in character, an interim result to 
be used until data on a more uniform plan could be 
obtained ; indeed the complete charting of the heavens by 
J. Franklin-Adams was largely inspired by the influence of 
Kapteyn, jointly with Sir David Gill, for the purpose of 
obtaining more satisfactory statistics. Yet the great 



192 STELLAR MOVEMENTS CHAP. 

divergence between the old and the new is surprising, and 
it may be well to attempt to trace the precise source from 
which it arises. 

In the first place it must be remarked that Kapteyn's 
figures for 17'"'0 are an extrapolation ; his data did 
not go beyond 14 IU *0. If then we compare the results 
for 14 m< 0, we have to account for a discordance- 
Ratio Star-density, Zone I to Zone VIII Kapteyn II 1 5 : 1 
,, ,, ,, ,, Chapman and Melotte 3 '9 : 1. 

Kapteyn made use of seven main sources of information. 
Of these the first four (including counts on the plates of 
the Cape Photographic Durchmusterung) relate to stars 
brighter than 9 m> 25 ; these show no important divergence 
in galactic distribution from the new fi (Hires. For the 

O o 

fainter stars the main reliance was placed on Sir John 
Herschel's star-gauges, 3 i.e., counts of stars visible with his 
18-inch reflector in a field of definite area. Although these 
are confined to the southern hemisphere, they are more 
evenly distributed and more typical of normal parts of the 
heavens than those of Sir William Herschel, and are there- 
fore to be preferred. Arranged according to galactic 
latitude these gauges give a star-density falling steadily 
from 1375 stars per square degree at the galactic circle to 
137 at the galactic pole. The limiting magnitude is 
determined by indirect means to be 13 '9, and it will be 
seen that the ratio 10 : 1 is practically equivalent to the 
definitive result adopted by Kapteyn. 

It appears, then, that the Herschel gauges are the main 
source of the discrepancy; but the results were not accepted 
without careful checks being applied. These were of two 
kinds ; first, counts of stars were made on forty-five photo- 
graphs, chiefly of the fields of variable stars, for which the 
limiting magnitude on a visual scale could be calculated 
from standard stars, photometrically determined. These, 
when arranged according to galactic latitude, gave results 



ix COUNTS OF STARS 193 

in excellent accordance with the star-gauges ; and it was 
considered that this checked the constancy of Herschel's 
limiting magnitude. The remaining source of statistics 

o o o 

was provided by the published charts of the Carte du del 
taken at Algiers, Paris, and Bordeaux. There is no means 
as yet of determining the limiting magnitude of these 
independently ; but, as it seems reasonable to assume 
that any fluctuations will be accidental and have no 
systematic relation to galactic latitude, they may be used 
to obtain the ratio of galactic concentration. The ratio 
found by Kapteyn between the belts of latitude 20 C 
and 40" -90 was 5'5 : 1. For Zones I and VIII the 
ratio would naturally be greater, and, moreover, the 
numbers refer to a limit about one magnitude brighter 
than Herschel's gauges. The ratio 10 : 1 at magnitude 
14 is therefore supported by three independent sources of 
evidence Sir J. Herschel's gauges, counts on variable 
star fields, and counts on the French astrographic charts. 

It has been suggested by H. H. Turner * that the 
discordance is due to a real difference in the distribution 
according as the magnitudes are reckoned visually or 
photographically. The Herschel counts refer directly to 
visual magnitudes, and the counts of variable star fields, 
although made on photographs, are reduced in such a way 
that the results refer to the visual scale. The counts on 
the French chart-plates, however, relate solely to the 
photographic scale ; and it is only by disregarding this 
evidence that the suggestion reconciles the two results. 
Perhaps we may consider the third source more doubtful 
than the two former, for the plates are distributed only 
through a narrow zone, and may be affected by abnormal 
regions of the sky. The possibility of a real difference 
in the galactic concentration for visual and photographic 
results is an interesting one. It appears to mean that 

* Who had also arrived at a small value of the galactic concentration for 
photographic magnitudes (Monthly Notices, Vol. 72, p. 700). 

O 



i 9 4 STELLAR MOVEMENTS CHAP. 

there are in the galactic regions vast numbers of faint 
stars too red to be shown on the photographs. This may 
be due either to a special abundance of late type stars in 
the more distant parts of the stellar system, or more 
probably to the presence of absorbing material a fog in 
interstellar space, which scatters the light of short wave- 
length. 

The hypothesis requires further confirmation. A dis- 
cussion by E. C. Pickering is directly opposed to it. He 
also used the visual determinations of magnitude in the 
fields of variable stars for his data ; but he applied them 
more directly. His conclusion was that " the number of 
stars for a given area in the Milky Way is about twice as 
great as in the other regions and the ratio does not increase 
for faint stars down to the twelfth magnitude 4 ." 

One of the most interesting results of Chapman and 
Melotte's investigation is that the total number of stars in 
the sky for the fainter magnitudes is much smaller than 
has often been supposed. Kapteyn's table gave 389 
million down to 17 m> against 55 million according to the 
present investigation. The excess of Kapteyn's numbers 
is almost wholly due to his high value of the galactic 
concentration ; the two investigations are practically in 
agreement at the galactic poles. 

By inspection of Table 32 it will be seen that the rate 
of increase in the total number of stars has fallen off very 
considerably for the last few magnitudes included. It 
appears that the numbers are beginning to approach a 
limit. An attempt to determine this limit involves a 
somewhat risky extrapolation, yet the convergence has 
already become sufficiently marked to render such extra- 
polation not altogether unjustifiable. The empirical 
formula \ogB m = a + 0m ym 2 cannot be pressed far 
beyond the range for which it was determined, since it leads 
to the impossible result that the number of stars down to a 



ix COUNTS OF STARS 195 

given magnitude would ultimately begin to diminish. By 
a simple modification more suitable formulae can be 

1 T> 

obtained. Instead of using B m we consider b m = - 

dm 

that is to say, we use the number of stars of the magnitude 
m instead of the number brighter than m. It is found 
that an equally good approximation is obtained by setting 

log ]0 6 )H = a + bm -cm 2 ......... (2) 

and with this form the whole number of stars approaches 
asymptotically to a definite limit as m is increased. 
We have then 



dm 



(a+bm-cml),log u e 



Cm , 

B, n = / 6 din 

, ry*-> ^ ....... 

Vfl" J 'JO 

where 



A = /" logic*? m 10 a-f 62 /4c B = / _C_ Q = _& 

V c V Iog 10 e 2c 

From the formula (3) it is seen that A represents the 
total number of stars of all magnitudes, and (7 represents 
the median magnitude, i.e., the limit to which we must 
go to include half the stars. 

As c is not very easy to determine, two formulae may 
be given between which the truth probably lies : 

Iog 10 6 m = -0-18 +0720 m -O'OlGOm 2 . . (4) 
Iog 10 &m = +0-01 +0-680 m -0'0140m 2 . . (5) 

We have here altered the unit of area, so that b m refers 
to the number of stars in the whole sky instead of to a 
square degree. 

These formulae lead to 

(4) (5) 

Whole number of stars of all magnitudes 770 millions 1800 millions 
Median magnitude .......... 22 m '5 24 m> 3 

Chapman and Melotte conclude : " Unless the general 
form of our expression for B m ceases to apply for values 

o 2 



196 STELLAR MOVEMENTS CHAP. 

of m greater than 17 (up to which the accordance is good) 
it is possible to say with some probability that half the 
total number of stars are brighter than the 23rd or 24th 
magnitude, and that the total number of stars is not less 
than one thousand millions and cannot greatly exceed 
twice this amount/' 

The mean star-density for given galactic latitude and 
limiting magnitude has been given in Table 31. A 
question arises as to how far these are sufficient to determine 
the star-density at any particular spot, and what variations 
from the mean are likely to arise. The possible variations 
may be classed as follows. 

(1) A systematic difference between the north and south 
galactic hemispheres. 

(2) A systematic dependence on galactic longitude in 
some of the zones. 

(3) General irregularity. 

There is not a very conspicuous difference between the 
two galactic hemispheres. So far as the tenth magnitude 
the southern hemisphere is found to be the richer by 10 or 
15 per cent. ; for fainter stars no difference is found. We 
have to depend for this conclusion on researches made 
before the introduction of modern standard magnitudes, 
but the evidence seems to be satisfactory. To account for 
the small difference in richness it has been usual to suppose 
that the Sun is a little north of the central plane of the 
stellar system ; this agrees with the appearance of the 
Milky Way, which deviates slightly from a great-circle, 
and has a mean S. Galactic Latitude of 17, 

It is the opinion of most investigators that, except in 
the Milky Way itself, the differences depending on 
galactic longitude are inconsiderable ; galactic latitude 
is the one important factor and overshadows all other 
variations. This view seems to be l>;is<>l on ;i ^cimral 
impression rather than on any quantitative results. A 



ix COUNTS OF STARS 197 

detailed investigation is greatly to be desired, for the 
conclusion must still be considered open to question. The 
following calculation appears to furnish an upper limit 
to the possible variations of the limiting magnitude. 

For the majority of the Franklin- Adams chart-plates, 
unstandardised counts have been given by Chapman and 
Melotte. Taking, for example, the Johannesburg plates, 
the centres of which lie between galactic latitudes 20 29 
and 30 39 , we find the following results, which refer to 
a limiting magnitude about 17 ni '5 : 

Zone. 20 2 29 30 39 

Number of Plates 20 16 

Smallest number of stars per plate . . 292,000 . 306,000 

Largest ,, . . .737,000 577,000 

Average deviation of log density from 

the mean 0'078 0'059 

Corresponding ratio 1*20 : 1 1'15 : 1 

The average deviation (20 and 15 per cent, respectively) 
includes not only the actual fluctuations of star-density, 
but variations due to the quality of the plates and the 
personality of the counters. Its smallness bears witness 
to the uniformity of the Johannesburg sky as well as 
to the regularity of stellar distribution. 

There is no doubt that within the limits of the Milky 
Way very considerable variations of star-density occur. 
The most remarkable region is in the constellation Sagit- 
tarius, where certain of the star-clouds are extraordinarily 
rich. This part of the Milky Way is unfavourably placed 
for observation in the latitude of the British Isles ; but 
from more southerly stations it appears the most striking 
feature of the heavens. It was found that on the 
Franklin-Adams plates the images of the faintest stars 
in this region were so close as to merge into a continuous 
background and it was impossible to count them. We 
should expect that the presence of the Milky Way 
clusters would add great numbers of stars beyond the 
normal increase towards the galactic plane, and it is 



198 STELLAR MOVEMENTS CHAP. 

rather surprising that there is not a more marked dis- 
continuity between the numbers for Zone I and Zone II 
in Table 31. Probably the dark spaces and tracts of 
absorbing matter, which are a feature of the Milky Way, 
neutralise the effect of the rich regions, and bring about a 
general balance. 

The star-ratio falls a great deal below its theoretical 
value 3 '9 8 for an infinite universe even as early as the 
sixth magnitude. The magnitude-counts are not, however, 
sufficient by themselves to determine the rate at which 
the stars thin out at great distances. For this we need 
additional statistics of a different kind, as will be shown 
in the next chapter. Meanwhile, although the star-counts 
do not determine a definite stellar distribution, we can 
examine whether any simple form of the law of stellar 
density in space is consistent with them. The simple 
result that the galactic concentration is independent of 
the magnitude, even if it were rigorously true, would not 
admit of any correspondingly simple interpretation in 
terms of the true distribution in space. It is therefore of 
no help in our discussion. 

Consider a stellar system in which the surfaces of equal 
density are similar and similarly situated with respect 
to the Sun as centre, the density falling off from the inner 
parts to the outer. We naturally think of spheroids of 
the same oblateness. 

Let the ratio of the radii towards the galactic pole and 
in the galactic plane be 1 : v. 

Corresponding to an element of volume S at a distance 
r towards the pole, there will be an element of volume v 3 S 
at a distance vr in the galactic plane, containing stars 
distributed with the same density. The number of these 
stars, being simply proportional to the volume, will be ^ 
times as many in the second case, and this will apply to 
all grades of intrinsic brightness separately. But their 



ix COUNTS OF STARS 199 

apparent brightness will be diminished in the ratio v~- 9 or, 
expressed in magnitudes, they will be 5 log v magnitudes 
fainter. This holds for all the elements of volume S in a 
cone from the Sun to the galactic pole, and the correspond- 
ing elements v s S in a cone from the Sun to the galactic 
equator. 

Hence if the number of stars brighter than a given 
magnitude is given by 

B m = fy(m) for the galactic pole 

it will be given by 

B m = v 3 \lr(m - 5 Iog 10 v) for the galactic plane. 

Now we have found (Table 33) that for the galactic 
pole 

\oB m = 0-901 +0-380 (m- 11) -0'013 (m-11) 2 . 

Hence for the galactic plane 

logB m = 31ogx +0-901 -0-380 x51ogi/ -0'013 (5 log*)- 

+ (0-380 + 0-013 x 10 log v) (m - 11) -0'013 (i - II) 2 . 

If log v = 0'54, this reduces to 

log# w = 1-400 +0-450 (in- 11) -0'0130 (m- II) 2 , 

which is fairly close to the true value for Zone I, viz. 

log.B m = 1-404 +0409(m-ll) -0-0139 (MI- 11)-. 

The excess of the calculated coefficient 0*450 over its 
observed value can be interpreted to mean that the 
diminution of density in the galactic plane is rather more 
rapid than would be the case if the surfaces of equal 
density were similar. The oblateness of the distribution 
becomes less pronounced at great distances. The differ- 
ence is not due to accidental error, for it can be shown 
that any other pair of zones would have given a greater 
excess in the same sense. The oblateness v is therefore 
not a constant quantity ; and the above value corresponds 
to a certain average distance which may be roughly 
expressed as the distance of the stars of the eleventh 
magnitude. 



200 STELLAR MOVEMENTS CH. ix 

For logi> = 0'54, we have v = 3'5. This, then, is the 
average oblateness, or ratio of the axes of the surfaces of 
equal density, v must not be confused with the galactic 
concentration of the stars ; it is a pure accident that their 
values are nearly the same. 

REFERENCES. CHAPTER IX. 

1. Chapman and Melotte, Memoirs, R.A.S., Vol. 60, Pt. 4. 

2. Kapteyn, Groningen Publications, No. 18. 

3. J. Herschel, Results of Astronomical Observations at the Cape of Good 
Hope, Cp. iv. 

4. Pickering, Harvard Annals, Vol. 48, p. 185. 



CHAPTER X 

GENERAL STATISTICAL INVESTIGATIONS 

IN the application of mathematics to the study of 
natural phenomena, it is necessary to treat, not the actual 
objects of nature, but idealised systems with a few well- 
defined properties. It is a matter for the judgment of the 
investigator, which of the natural properties shall be 
retained in his ideal problem, and which shall be cast 
aside as unimportant details ; he is seldom able to give a 
strict proof that the things he neglects are unessential, but 
by a kind of instinct or by gradual experience he decides 
(sometimes erroneously it may be) how far his representa- 
tion is sufficient. 

In the idealised stellar system, which will now be 
considered, there are three chief properties or laws. The 
determination of these must be regarded as the principal 
aim of investigations into the structure of the sidereal 
universe, for if they were thoroughly known we might 
claim a very fair knowledge of the distribution and move- 
ments of the stars. The laws are : 

(1) The density law. The number of stars per unit 
volume of space in different parts of the system. 

(2) The luminosity law. The proportion of stars 
between different limits of absolute brightness. 

(3) The velocity law. The proportion of stars having 
linear velocities between different limits both of 
amount and direction. 

201 



202 STELLAR MOVEMENTS CHAP. 

As regards the first of these, the density may be assumed 
to depend on the distance from the Sun and on the galactic 
latitude. The decrease of density at great distances from 
the Sun represents the fact that the stellar system is 
limited in extent, and as it is notorious that the limits are 
very much nearer towards the galactic poles than in the 
galactic plane, a representation which did not include a 
variation with galactic latitude would be very imperfect. 

The luminosity law and velocity law may be assumed 
tentatively to be the same in all parts of space. Argu- 
ments may be urged against both these assumptions ; but 
they seem to be inevitable in the present state of knowledge, 
and it is probable that the results obtained on this basis 
will be valid as a first approximation. It may further be 
remarked that in dealing with proper motions we are 
necessarily confined to a rather small volume of the 
stellar system, and the assumption of a constant velocity 
law in such investigations seems to be justified. 

The constancy of the velocity law is in most investiga- 
tions assumed in a different form, which must be carefully 
distinguished from the assumption just stated, and is, in 
fact, much less innocuous. It is assumed that for the 
stars of a catalogue the velocity law is the same at all 
distances. Now among the stars of a catalogue with a 
lower limit of magnitude there is a strong correlation 
between luminosity and distance. Thus an additional 
assumption is virtually made, that the stars of different 
intrinsic luminosity have the same velocity law ; or, since 
spectral type and brightness are closely associated, that 
the stars of different spectra have the same velocity 
law. This is well-known to be untrue. It seems likely 
that results obtained on this assumption (including some 
investigations in this chapter) may be misleading in some 
particulars ; though here again we may often arrive 
at fairly correct conclusions notwithstanding imperfect 
methods. But it is desirable, where possible, that the 



x STATISTICAL INVESTIGATIONS 203 

different spectral types should be investigated separately : 
for in the case of stars of homogeneous type we know of no 
evidence to invalidate the hypothesis of a constant velocity 
law. 

A further property of the stellar system, which has some 
claim to be retained in the -idealised representation, is the 
absorption of light in inter-stellar space. There is some 
evidence, insufficient it may be, that this is small enough 
to be neglected in the present discussions. As it does not 
seem practicable to deduce useful results when it is retained 
as an unknown, we shall take the risk of neglecting it. 

The problem of determining one or more of the three 
laws that have been enumerated may be attacked in various 
ways ; and the diversity of the investigations is rather 
bewildering. The difficulty of giving a connected account 
of the present state of the problem is increased by the 
fact that some of the work has been based on data that are 
now obsolescent ; and it is difficult to know how far the 
introduction of more recent figures would produce import- 
ant modifications. 

The general statistical researches described in this 
chapter depend on one or more of the following classes 
of data : 

(a) Counts of stars between given limits of magnitude. 

(b) The mean parallactic motion of stars of given mag- 
nitudes. 

(c) Parallaxes measured directly. 

(d) The observed distribution (or spread) of the proper 
motions of stars brighter than a limiting magnitude. 

The radial velocities are only appealed to for the adopted 
speed of the solar motion, usually taken as 19*5 km. 
per sec. 

It is convenient to distinguish the use of the proper 
motions for determining the mean parallactic motion from 
other applications of the proper motion data. The paral- 
lactic motion, or as it is sometimes called, the secular 



204 STELLAR MOVEMENTS CHAP. 

parallax, fixes the mean distance of a class of stars 
without introducing any consideration of the distribution 
of their individual velocities. 

The investigations may be grouped into three classes : 

I. Those which depend on (a) and (b) only. 

II. Those which depend on (a), (6), (c), and (d). 

III. Those which depend on (d) only. 

It will be shown later that (a) and (b) are theoretically 
sufficient to determine the density and luminosity laws, 
so that the inclusion of (c) introduces some, redund- 
ancy of equations. Investigations depending on (d) stand 
rather apart from the others, since they involve the velocity 
law ; but as they also throw some light on the other two 
laws, it is useful to consider them in the same connection. 

Before proceeding to consider the three classes of 
investigation, it is necessary to give some attention to the 
expression of the results for the mean parallactic motions 
and the measured parallaxes in a suitable form. The 
formulae given by J. C. Kapteyn in 1901 are very widely 
used ; and, although it would naturally be an improve- 
ment to substitute more recent data, no general revision 
of his work has yet been made. Kapteyn derived two 
formulae 'for the mean parallaxes of stars ; one 
expressing the mean for all stars of a given magnitude, 
the other for stars of given magnitude and proper motion. 
For the first formula it is not possible to make use of the 
directly measured parallaxes. Jt is not so much that the 
data are too scanty, as that the stars are usually selected 
for investigation on account of large proper motion, and 
are accordingly much nearer to the Sun than the bulk of 
the stars of the same magnitude. The mean parallactic 
motion, or secular parallax, provides the necessary means 
of determining the dependence of parallax on magnitude 
alone. The only practical difficulty arises from the 
occasional excessive motions, which exercise a dispropor- 



STATISTICAL INVESTIGATIONS 



205 



tionately great influence on the result. Kapteyn's results, 
which depend on the Auwers-Bradley proper motions, are 
contained in the formula, 



Mean parallax for magnitude m = 0""0158 x 



(A) 



If Types I and II are taken separately, the formula? 
for the mean parallaxes are : 

Type I 7? m - 0"'0097 x (078)-5-5. 

,, II 7f w = 0"-0227 x (078)-5'5. 

In Table 34 the mean parallaxes for the different 
magnitudes are given, after correcting the foregoing 
formula (A) to reduce to the modern value of the solar 
motion, 19*5 km. per sec., instead of 16*7 used by 
Kapteyn. 

TABLE 34. 

I\<i/>teyns Mean Parallaxes. 
(Reduced to the value 19*5 km. per sec. for the solar motion.) 



Magnitude. 


Mean Parallax. 


Magnitude. 


Mean Parallax. 


1-0 


0-0414 


6-0 


0-0120 


2-0 


0-0323 


7-0 


0-0093 


3-0 


0-0252 


8-0 


0-0073 


4-0 


0-0196 


9-0 


0-0057 


5-0 


0-0153 







For the dependence of parallax on proper motion, 
Kapteyn had recourse to the measured parallaxes. For 
this purpose their use is quite legitimate, though we may 
be inclined to doubt whether the data (at that time much 
less satisfactory than now) were sufficiently trustworthy. 
For a constant magnitude, the dependence on the proper 
motion p was found to follow the empirical formula 



where _p=l,r405. The actual formula giving the 



206 STELLAR MOVEMENTS CHAP. 

mean parallax of a star of magnitude in and proper 
motion rf' per annum is 

tm,n. = (0-906)-5-5 x (0-0387/i)' n2 ' (B) 

It is interesting to know how nearly this mean formula 
is likely to give the correct parallax of a particular star. 
Assuming that log(7r/7r) is distributed according to the law 
of errors, the probable deviation of this logarithm has 
been found to be 0'19. Thus it is an even chance that the 
parallax of any star will lie between 0'65 and 1*55 times the 
most probable value for the given motion and magnitude.* 
This result was found by comparing measured parallaxes 
with the formula, and determining the average residual. 

The formulae for 7r m and ir m ^ may be more conveniently 
expressed in the form t 

Iog 10 irm = - M08 -0-125m, (C) 

log 10 7r, M = - 0-766 -0'0434m +0-712 log /x ' (D) 

In order to bring together all the data due to Kapteyn, 
we may add here his results for the counts of stars of 
successive magnitudes. It has been shown by K. Schwarz- 
schild that Kapteyn's numbers can be summarised by the 
formula : 

Iog 10 6, n = 0-596 + 0-5612 m- 0-0055 m 2 (E) 



where b m is the number of stars (in the whole sky) between 
magnitude m and m +dm. 

The formulae (C), (D), and (E) correspond respectively 
to the data (6), (c), and (a) previously mentioned. 

The main criticism, of subsequent investigators has 
been directed against the large value of the coefficient of 
m in formula (C). There is some reason to believe that 
the decrease of mean parallax with increasing faintness is 
less rapid than is shown in Table 34. According to C. V. 

* The most probable value is not the mean value. With the above prob- 
able deviation the most probable value would be 0'81 x 7r m , M . 

f Formula (A) and (C) do not quite correspond as the former contains a 
slight correction made by Kapteyn (Preface, Groninyen Publications, No. 8). 



x STATISTICAL INVESTIGATIONS 207 

L. Oharlier 1 the coefficient is less than half Kapteyn's 
value. Charlier's determination rests on the Boss proper 
motions, which are of great accuracy ; but the range of 
magnitude is too limited for a satisfactory solution. 
Without attaching much weight to the precise value, he 
considers that Boss's proper motions cannot be reconciled 
with the larger figure. This means that there is less 
difference between the parallaxes of different magnitudes 
than Kapteyn supposed. The writer, 2 also working on 
Boss's Catalogue, had found that for stars of a given 
magnitude the spread in distance must be less than that 
deduced from Kapteyn's formulae, a fact which is evidently 
related to Charlier's objection. 

G. C. Comstock 3 has likewise maintained that the paral- 
laxes of faint stars are larger than those given by the 
formula. From an investigation of the proper motions of 
479 stars from 7 m to 13 m he concluded that a relation, 
first given by A. Auwers for a shorter range, holds satis- 
factorily from the third to the thirteenth magnitudes, 
viz., that the mean proper motion is inversely proportional 
to the magnitude. As the mean proper motion may be 
taken to be proportional to the parallax, Comstock's result 
leads to the formula 

TT = c/m (in >3). 

The formula (E) may be compared with Chapman and 
Melotte's determination (formulae (4) and (5), p. 195). 
The real differences are very large, though perhaps not 
quite so great as would appear from a cursory comparison 
of the coefficients. 

I. INVESTIGATIONS WHICH DEPEND ON COUNTS OF STARS 
AND ON MEAN PARALLACTIC MOTIONS. 

From the mean parallaxes of stars of different magni- 
tudes combined with the counts of the number of stars 
down to limiting magnitudes, the density law and the 



208 STELLAR MOVEMENTS CHAP. 

luminosity law can be determined. A most elegant gene- 
ral solution of this problem has been given by Schwarz- 
schild ; and, although it may usually be less laborious in 
practice to work out special cases according to the func- 
tions which represent the observed data, his method is so 
generally applicable to the fundamental problems of 
stellar statistics that we shall begin by considering it at 
length. 

Let D(r) be the number of stars per unit volume at a 
distance r from the Sun. 

Let $(i) di be the proportion of which the absolute 
luminosity lies between i and i + di. 

Setting h for the apparent brightness of a star, we have 



Let B(h) dh be the total number of stars of apparent 
brightness between h and h + dh. 

And let ir(h) be the mean parallax of stars of apparent 
brightness h. 

o 

Then the whole number of stars at distances between 
r and r + dr is 

. D(r), 



and of these the proportion <(Ar 2 ) r 2 dh will have an 
apparent brightness between h and h + dh. 
Hence 



H(h)dh = 
or 

B(h) = 

And for the sum of the reciprocals of the distances 



We now transform the two integrals (2) and (:*) as 
follows : 



x STATISTICAL INVESTIGATIONS 209 

Let 

r = e ~ P, h = e- 2 M, 

so that 

i = e ~ 20* + P). 

Further let 



60*), 



Here, since .!?(/&) and TT(^) are supposed to be given by 
the data of observation, b(fi) and c(^) are given likewise ; 
they are the same observed quantities expressed as func- 
tions of a changed independent variable. 

The two integral equations (2) and (3) become 



&GO= f(p)9(t*+p)dp ................ (4) 

J -x 

c( M )= / f(p)g( f j L +p)e P dp ............... (5) 

.' -30 

Let us form the Fourier integrals 

*() = -i- r 

5!rJ -oo 



K) = -I/'" / M )e-.^ ( )-- L|" flOOe-*da 

27T.' -ao 27JV -oo 

where t= ^/ 1. 

Then we have the well-known reciprocal relation 

J -00 J - 9 

. x J - 

Now 



i /" x / x 

= o-/ / 

7TJ -x J - 

= ^/" f(p 

-STTj - x 

Hence by (6) 



. . (8) 
P 



eiqp 

-oo 



210 STELLAR MOVEMENTS CHAP. 

Similarly, 

-j / 

<(</) = o- I 

^TJV -o 

= 9(9) r 

J -x 

Thus 

cfa) = 2irf(i- 9)9(9) ................... (9) 

From (8) and (9) we have 

fi=i> = <i> . (10) 

!(-) K) 

As the functions c and fc can be calculated directly (by 
Fourier analysis or otherwise) from c and b, the right- 
hand side is known. 

Setting 

p = iq 

and 



equation (10) becomes. 

F(^ + l)-^) = log^|) .............. (11) 

a difference equation of which the solution is 



the integration being along the imaginary axis of p'. 

Thus F can be found, and from it f. Then f is deter- 
mined by means of (7). 

Further, when f has been determined, g is given by 
(8) and then g can be determined. 

The density and luminosity laws are accordingly found. 

If particular forms are assumed for the expressions 
which give the counts of stars of each magnitude and their 
mean parallaxes, the analysis may be made much simpler. 
In the following investigation special forms of the functions 
are adopted, which appear adequate to represent the present 



x STATISTICAL INVESTIGATIONS 211 

state of our knowledge, and lend themselves conveniently 
to mathematical treatment : 
Let 

r be the distance measured in parsecs, 

/ the absolute luminosity measured in terms of a star of zero magnitude 

at a distance of one parsec , 
h the apparent brightness in terms of a star pf zero-magnitude. 

And set 

p = -5'01og 10 r, 
M= -2-51og 10 i, 
m = 2*5 Iog 10 #, 

where M and m are accordingly the absolute and apparent 
magnitudes. 

We have i = hr 2 , and M = m + p. 

We adopt the forms : 

Density law: D(r) = 10o - "IP - 2P 2 .......... (13) 

Luminosity law : </>(i) = 10 ft o - &i-V - &2-V2 ......... (14) 

Number of stars between magnitudes m and m + dm : 
6(m)(Jm = 10- !-** dm ........ (15) 

Mean parallax of stars of magnitude m 

7r(r/i) = 10 p o--Pim-P 2 m2 ......... (16) 

Then, expressing that the stars of magnitude m to 
m + dm are made up from successive spherical shells at 
distance r, containing 4?ir 2 dr D(r) stars, of which the 
proportion < (/*/'") /' 2 dh are of the appropriate intrinsic 
brightness, we have 



(17) 

Now 

dm = - 2'5 Iog 10 e dh h, 
dp = -5'01og 10 e drjr, 

r = 10- 0"2 P 



and 

(r) = 10&o - *i(m + P) - Wm + p)2 + oo - o 1P - aapS^ 

p 2 



212 STELLAR MOVEMENTS 

Hence we have b(m) = 

dp . 10-p-' 4Bl + 0-lP-2P 2 + 



CHAP. 



. . (18) 

12-5 (log e) 2 J -a, 

Now the integral is of a well-known form, 

r dp !CMo- 

J -oo 

The reduction evidently leads to an expression for b(m) 
of the form set down in (15), and we find 



(19) 



The sum of the parallaxes of stars between magnitudes 
ra and m + dm, which is equal to ir(m) b(m)dm, is found 
by writing r z for r* in (17), or ir(m) b(m) is given by writing 
^-0*2 for a^ in (18) and (19). Carrying through this 
change in (19), we find 



(20) 



+P _ 



Hence subtracting (19) from (20) 




(21) 



The fact, that quadratic formulae for the logarithms of 
the density and luminosity functions lead to a linear 
formula (P 2 = 0) for the logarithm of the mean parallax 
is of interest, because the linear formula for the latter 
is the one given by Kapteyn, and is in general use. 

From the formulae (19) and (21) it is easy to deduce the 



x STATISTICAL INVESTIGATIONS 213 

coefficients of the density and luminosity functions in 
terms of the observed coefficients * , K l9 * 2 , P , and P lt 

Thus a 2 = I, , If - = :. , and so on. a and b are not 
O/i 1 



given independently, but as a sum a + 6 . But, if desired, 
6 can be found from the condition implied in the definition 
of 



Of practical attempts to determine the density and 
luminosity functions from the mean parallaxes and 
counts of stars of different magnitudes, an investiga- 
tion by H. Seeliger 4 (1912) may be taken as .an example. 
Dividing the sky into five zones according to galactic 
latitude, he obtained for B, n (the number of stars per 
square degree down to magnitude m) the following 
expressions : 

Zone. Galactic Latitude. Formula. 

A 90" to 70 log lo B Ifl = -4-610 +0*6640 m - '01334 m 2 

B 70 c to50 -4-423 + 0'6099 m - "00957 m 2 

C 50 : to30 ,, -4-565 +0'6457 m - '01025 m 2 

D 30= to 10 = -4-623 +0'6753 m -0-01027 m 2 

E + 10' to- 10 ' ,, -4-270 +0-6041 m -0 '00512 m 2 

These were derived from a discussion of the magnitudes 
of the Bonn Durchmusterung and from Sir John Herschel's 
gauges. Owing to the use of the latter source, the 
galactic concentration of the faint stars is very strong, 
just as in Kapteyn's results ; the doubt cast on these 
numbers by modern researches has been fully discussed in 
the preceding chapter. The course of Seeliger's coefficients 
from zone to zone appears quite irregular, but the form of 
the expressions tends to conceal a steady progression in 
the actual numbers. 

From these expressions for B m , b m = ^ was deduced 

dm 

without difficulty and expressed in the same quadratic 
form (equation 15). For the mean parallaxes (equation 



2i 4 STELLAR MOVEMENTS CHAP. 

16) Kapteyn's numbers were adopted. But a difficulty 
arises because the meau parallaxes have been given only 
for the sky as a whole, and not for the separate zones of 
galactic latitude ; and it is well known that the mean 
parallax varies greatly from one latitude to another. The 
difficulty, however, can be surmounted. We have agreed 
to regard the density law as depending on the galactic 
latitude, and the luminosity law as constant ; that is to 
say, the coefficients , a lt a 2 will be different for each 
zone ; b , b lt b. 2 will be the same for all. If we eliminate 
! and a 2 from (19) and (21), and disregard the first 
equation, which is only used for determining a + 6 , we 
obtain two equations between b ly b. 2 and P , P 1 from each 
zone. Combining these according to the number of stars 
in each zone, two equations are obtained in which it is 
permissible to substitute the mean P l and P given by 
Kapteyn's parallaxes. These determine b l and b. 2 . The 
remaining constants a + b , a lt a 2 are then determined for 
each zone separately by means of equations (19). By a 
procedure of this kind, though differing in detail, Seeliger 
arrived at a solution. That the result is very different 
from what would be obtained by using the same parallax 
formula for all zones may be seen by the following 
numbers, which were deduced as the mean parallaxes of 
stars of magnitude 9 m< 0. 

Zone. A. B. C. D. E. 

Mean Parallax (9 m '0) (T0065 0"0061 CT0053 0"'0044 0"'0039 

The density and luminosity functions are given by (13) 
and (14). Seeliger's result for the luminosity function is 

#t) = const, x*- 2 ' 12910 ^- ' 1007 ' 10 *'* (22) 

As examples of the law of density the following (given 
iv Seeliger) may suffice: 

Zone. A. B. 0. D. E. 

Density at distance 5000 parsecs Q . mi ^^ Q . 0355 

Density at the bun 
Density at 1600 parsecs . Q21 ^^ . 1Q7 

Density at 10 parsecs 



x STATISTICAL INVESTIGATIONS 215 

These results show a much more rapid falling off in 
density near the poles compared with the galactic plane. 
The results for Zone E should perhaps be set aside, since 
presumably they are disturbed by the passage of the Milky 
Way through that region ; but the other four zones repre- 
sent the general distribution in the stellar system. We 
do not, however, place much reliance on the numerical 
results, since the determination rests on the Herschel 
gauges and on Kapteyn's mean parallaxes, both of which 
are open to some doubt. 



II. INVESTIGATIONS WHICH DEPEND ON COUNTS OF 
STARS, THE MEAN PARALLACTIC MOTIONS, MEASURED 
PARALLAXES AND DISTRIBUTION OF PROPER MOTIONS. 

Suppose that a table of double-entry is formed giving 
the number of stars between given limits of magnitude 
and given limits of proper motion. For the stars in any 
compartment of the table corresponding to magnitude ra 
and proper motion p, the formulae (B) or (D) give the 
mean parallax. Further, as already stated, the individual 
parallaxes deviate from the mean according to the law, 

Frequency of log (TT/TT) is an error function with probable error 0'19. 

Hence the proportion of these stars between any 
given limits of parallax can be found. Thus, taking 
the stars in any one compartment, we can redistribute 
them into a new table with arguments parallax and 
magnitude. Treating each compartment of the old table 
separately, we transfer all the stars into the new table, 
and obtain the number of stars between given limits of 
magnitude and of parallax. 

Table 35 gives the results of a solution made in this 
way by J. C. Kapteyn. 5 It shows how the number of 
stars of each magnitude are distributed as regards dis- 
tance. 



216 



STELLAR MOVEMENTS 



CHAP. 



TABLE 35. 
in Distance of Stars of each Magnitude (Kapteyn). 



Limits of Mean 
Distance. Parallax. 


Number of Stars in the Sky. 


m-M. 


T> // 


m. 


m. 


m. 


m 


m. 1 m. 




Parsecs. 


2T,-3r. 


3-6-4-6 


4-6 -5-G 


5-6-6-6 


6-6-7-6 


7-6-8-6 


m. 


>1000 


0-6 


5-0 


25 


127 


703 


4840 




631-1000 0-00118 


2-0 


8'0 


42 


197 


871 


4590 


14-5 


398 -631 0-00187 


2'9 


19-6 


92 


369 


1466 


6050 


13-5 


251398 ; 0-00296 


9-4 


29-6 


151 


603 


2210 


7310 


12-5 


158251 0-00469 


14-7 51-0 


223 


815 2770 


8320 


11-5 


100158 0-00743 


19-6 


64-6 


256 


885 2760 


5830 


10-5 


63100 ! 0-0118 


22-8 


72-8 


240 


767 


2080 


4140 


9-5 


4063 0-0187 


21-3 


71-1 


190 


537 


1240 


2150 


8-5 


2540 0296 


17-2 


57-1 


130 


311 


579 


890 


7-5 


1625 0-0469 


11-8 


39-1 


71 


145 


235 


316 


6-5 


1016 0-0743 


6-5 


225 


34 


56 


84 


99 


5-5 


6-3-10 0-118 


3-2 


11-2 


14 


18 29 


30 


4-5 


06-3 


2-0 


7-0 


8 


11 14 


14 






For stars of known magnitude and distance the abso- 
lute magnitude M can be calculated. The number to be 
subtracted from the apparent magnitude m to give the 
absolute magnitude will be found in the last column of 
the table. The limits of distance were so chosen that 
the step from one line to the next corresponds to a change 
of one magnitude. 

Each line of Table 35 gives a determination of the 
luminosity law, for it exhibits the number of stars in a 
certain volume of space which have absolute magnitudes 
between given limits. By taking suitable means between 
the results from each line of the table, Kapteyn arrived 
at an expression for the luminosity law which may be 
written 



const x e - 1 ^ log* < - o-07-j (log, i)3 



(23) 



This may be compared with Seeliger's result (22). 

Again, if we start from any number on the left side of 
Table 35, and move diagonally upwards to the right, the 
successive numbers will all refer to stars of the same 



x STATISTICAL INVESTIGATIONS 217 

absolute magnitude. For example, starting from 17*2, we 
have the numbers 

17-2 7M 240 885 2770 7310 

referring to an absolute magnitude 4*4 (strictly 4*9 
to -3'9). 

Now these are the numbers of stars in a series of 
spherical shells, the volumes of which form a geometrical 
progression 

1 4-0 15-8 63-1 251 1000 

Whence by division the relative densities are 

17-2 17-8 15-2 14-0 11/0 7'3 

corresponding to distances of 

25-40 4063 63100 100158 158251 251398 parsecs. 

According to our hypotheses, the change of star density 
with distance will be shown equally whatever absolute 
magnitude is selected ; we shall thus obtain from Table 35 
a number of determinations of the density law. The 
following numbers will illustrate the character of the 
density law deduced by Kapteyn. 

Distance. Star-density. 

1-00 

50 parsecs 0'99 

135 ,, 0-86 

213 0-67 

540 0-30 

850 0-15 

It has been mentioned that in this second class of 
investigation, more data are employed than are necessary 
to give a solution. That is why we obtain from Table 35 
a number of separate determinations* of the luminosity 
and density laws instead of a single solution. Kapteyn 
has used the agreement of the separate determinations to 
defend the assumption that the absorption of light in 
space is not large. Schwarzschild has discussed the 
mutual consistency of the data analytically and has 
shown that the theoretical relations are well-satisfied. 



2i8 STELLAR MOVEMENTS CHAP. 

Thus he finds that the theoretical value of the probable 
error of log ir/ir is 0*22, compared with 0'19 adopted 
by Kapteyn from the observations. 

IH. INVESTIGATIONS WHICH DEPEND ONLY ON THE 
DISTRIBUTION OF THE PROPER MOTIONS OF STARS. 

Another class of statistical investigations depends 
entirely on the proper motions. In this it is convenient 
to introduce a new definition of the density law, viz., the 
number of stars brighter than a limiting apparent 
magnitude per unit volume of space at different distances 
from us. This involves a combination of the old density 
and luminosity laws, and in itself gives us no definite 
information as to the structure of the system ; but it is 
clearly a matter of great practical interest to know how 
the stars of our catalogues are distributed in regard to 
distance. The determination of the velocity law is also an 
object of these investigations. 

Let 

g(u) be the number of stars having linear motions between u and 
u 

u + duj 
h(a) the number having proper motions between a and a + da ; 

/(r) the proportion of stars (of the catalogue) at a distance between 
r and r + dr from the Sun. 

Then, expressing that h(a) ' l is made up of stars at all 

a 

possible distances r, with linear motions u between ra and 
r(a + da). 

*(.)*? = r&^V^rda. 
a J / ra 

Hence 



Or writing 



u = ey 



x STATISTICAL INVESTIGATIONS 219 

Equation (24) becomes 

IV =/" f(X) 8 (X+/i)dX ............. (25) 

J -oo 

This is of the same form as (4), and the solution is 
accordingly 

H(,j) = 2irF(-q)G(q) ............. (26) 

where F, G, H are the Fourier integrals corresponding to 



............. (27) 

&irJ oo 

and 

f(X) = T F(q)e^dq ............... (28) 



To obtain a starting point it is assumed that in the 
direction at right angles to the vertex of star-streaming 
the linear motions are distributed according to the error- 

_/ </ -( ? 4__ y\- 
law e ' du, where F is the component of the solar 

motion in that direction. The evidence for this has been 
discussed in Chapter VII. There is the advantage that 
both the two-drift and ellipsoidal theories agree on this 
point, so that it is a fair assumption in any comparison of 
the t\vo theories. Confining attention to the stars moving 
in this direction, g can be found, and hence G is deter- 
mined by periodogram analysis. For the same direction, 
H is determined by an analysis of the observed motions, 
and then F is given by (26). Having once found F 
(which determines the density law f), we may use its 
value in (26) to determine G for other directions, and 
g is determined from G by another periodogram analysis 
(equation (28)). 

Thus from the one assumption that the proper motions 
at right angles to the direction of star- streaming are 
distributed according to Maxwell's law, it is possible to 
determine the complete velocity-law. The method is a 
little difficult to apply, not only on account of the long 



22O 



STELLAR MOVEMENTS 



CHAP. 



numerical computations, but also because the formulae, 
which express statistical truths, cannot be exactly satisfied 
by observations depending on a limited sample of stars. 
The formal solution, which is unpleasantly conscientious, 
knows only one way out of such a difficulty ; it diverges. 
We therefore have to smooth the observed data before- 
hand so as to make sure that the task is not an impossible 
one ; and even then, if a trifling irregularity is left in, it 




Solar 
Antapex 



Scale 



20 30 40 50 Km. per sec. 



FIG. 21. Curves of Equal Frequency of Velocity. 

may cause the solution to make the most astounding 
oscillations in the attempt to follow it exactly. 

This method has been applied to 1129 stars, the proper 
motions being taken from Boss's Preliminary General 
Catalogue. The stars are in two opposite areas of the 
sky with centres at the two Equinoctial points. As these 
centres are nearly 90 from the vertex and from the solar 
apex, the star-streaming and solar motion are shown in 
this region without foreshortening. 

The resulting distribution of the linear velocities is 
shown in Fig. 21, which shows the curves of equal 
frequency. Setting the number of stars with component 



x STATISTICAL INVESTIGATIONS 221 

velocities between x and x + dx, y and y + dy proportional 
to ^(x,y) dxdy, the curves ^(x,y) = 1, 2, 3, etc., have been 
traced. In the neighbourhood of the origin the present 
method of calculating ty becomes indeterminate ; accordingly 
within a certain distance of the origin the equi-frequental 
lines have not been calculated, but the diagram has been 
completed by the dotted lines, for it is sufficiently obvious 
how the joining up must take place. "With the origin of 
co-ordinates shown in the figure the velocities are referred 
to the Sun ; they may be referred to any other standard by 
merely changing the origin. 

It will be seen that the diagram lends some support to 
the hypothesis of two distinct drifts as opposed to the 
unitary ellipsoidal hypothesis. It is not clear how far 
this indication of a division is to be trusted. I do not 
think that any modification of the analytical method or of 
the original assumptions would make any difference in 
the result ; but the data may be scarcely sufficient. 

As F(q) is determined in the course of the analysis we 
may also derive the distribution of the stars in regard to 
distance. The results are as follows : 



TABLE 36. 

Distribution of the Stars of Boss's Catalogue (for a mean Galactic 
Latitude 63). 



Distance. 


Parallax. Percentage of Stars. 


Parsecs. 







10-0 12-5 


0-lQ 0-08 


1-0 


12-5 167 


0-08 0-06 


1-5 


167 25-0 


0-06 0-04 


2-9 


25-0 50-0 


0-04 -0-02 


9-5 


50-0 667 


0-02 0-015 


8-6 


667 100 


0-015 O'OIO 


17-5 


100 125 


0-0100-008 


117 


125 - 167 


0-0080-006 


14-9 


167 - 250 


0-0060-004 


16-6 


250 -- 500 


0-004-0-002 


10-0 


500 1000 


0-0020-001 


2-4 



222 STELLAR MOVEMENTS CHAP, 

No less than 70 per cent, of these stars are between 
50 and 250 parsecs distant from us. It is an even chance 
that a 'star's parallax is contained within the limits 0"*012 
and 0"'004 ; in other words, if we know no more of a star 
than that it is brighter than 6 m '5, ive may set down iU 
parallax as 0"'008 0"'004.* 

In the foregoing investigation special attention wag 
devoted to the calculation of the velocity-distributioc 
with as little assumption as possible, and the distance -distri- 
bution was found rather as a bye-product. We shall now 
consider researches in which the leading object is to find 
the distance-distribution. 

Consider as usual the stars in an area of the sky 
sufficiently small to be treated as plane. 

Let the proper motions be resolved into components 
along the direction towards the vertex and at right angles tc 
it (in the plane of the sky). Denote the latter components 
by T). t 

Let 

- V= Component of the solar motion (in linear measure) in th< 

T} direction. 
h(r)) drj = Number of stars whose component rj of proper motion liei 

between 17 and rj + drj. 
g(v) dv = Proportion of stars whose corresponding component o 

linear motion lies between v and v + dv. 
/(r) dr = Number of stars (in the region) at a distance r to r + dr ; sc 

that the density of the stars brighter than the limiting 

magnitude at a distance r from the Sun is proportional t< 

r- 2 /(r). 

Let further 

H(rj) = \h(rf)dri = Number of stars whose component prope: 
J o 
motion is between and 17. 

f\r) = / f(r) dr'= Number of stars whose distance is less than r. 

* This remark is strictly true according to the ordinary definition o 
probable error, but in a skew distribution of frequency such as this th< 
probable error has not the pmjuTtirs ^ve usually associate with it. 

t The notation of the piwinu.*, investigation is not conveniently applicabl 
to this. We make an entirely fresh start. 



x STATISTICAL INVESTIGATIONS 223 

The units of r, v, and rj are made to correspond, so 
that 

i? = r;. 

It is not difficult to see that the number of proper 
motions greater than ij is to be found by multiplying the 
number of stars nearer than r by the proportion with 
linear velocities between rjr and v)(r + dr), for successive 
steps of r. Thus if JVj and JV 2 be the total numbers of 
stars with positive and negative components rj, we shall 
have 



and 

J^-JH-^ 

so that 

JTi+^i- {%)+!!( -*X-r FWg^Tjdr . . . (29) 

%/ ""00 

provided that we choose an even function to represent F(r). 
We assume that the rj components are distributed accord 
ing to the Maxwellian Law, so that 

0(r)= * e-W-r? . (30) 

VTT 

Consider now the special form : 






where k is a disposable constant. 
Substituting in (29) and setting 

rj = hi; 
T = hV, 

the equation reduces to 




say. 

The quantity T can be found from the ratio of the 
number of negative proper motions 17 to the whole number. 
This ratio is, in fact, 



i j" 

- 

S'TT J r 



22 4 



STELLAR MOVEMENTS 



CHAP. 



Thus the function R(ri) for the particular region can be 
tabulated. If then we can find a quantity k such that 
the proportion of stars having component proper motions 
numerically less than nk is represented by JR(n), then the 
distance distribution is given by (31). 

The form of f here discussed has been selected (by 
F. W. Dyson and the writer independently) from other 
forms lending themselves to mathematical treatment, as 
corresponding most nearly to the observed motions. We 
may make a second approximation by superposing two 
such functions with different values of k. 

Turning now to the results, we give first the distribu- 
tion of stars brighter than 5 m '8 (Harvard Scale) for Types 
A and K separately. It has been pointed out previously 
that the assumed law (30) is only valid when stars of 
homogeneous type are discussed. Two regions were 
investigated, one with centre at the equinoxes (Gal. 
Lat. 63), the other with centre at the poles (Gal. Lat. 27) ; 
the former was much the more favourable for the purpose. 

TABLE 37. 
Distribution of Stars brighter them 5 m '8. 



Limits of Distance. 


Type K. 


Type A. 


Parsecs. 


G. Lat. 63. 


G. Lat. 27. 


G. Lat. 63. 


G. Lat. 27.* 


25 


11-7 


9-2 


7'9 


7'5 


25 50 


30-7 


24-1 


' 18-2 


21-4 


50 75 


35-3 


31-5 


18-8 


327 


75-] 00 


31-4 


31-8 


17-6 


38-5 


100125 


24-8 


31-0 


167 


40-9 


125-150 


19-0 


287 


iu :; 39-0 


150175 


14-5 


26-5 


16-0 


34-5 


175200 


11-3 


23-3 


15-0 


27-9 


L'OO >.", 


8-2 


ID'S 


13-2 


21-3 


>r> 2r>0 


57 


15-6 


11-2 


15-3 


>250 


8-4 


36-3 


32-6 


25-0 



* North Polar area only. There appeared to be some abnormality near 
the South Pole. 



x STATISTICAL INVESTIGATIONS 225 

Each region occupies 0*235 of the whole sky, and the 
figures give the actual number of stars. 

The results for Type K, which appear to be the more 
trustworthy, show very well the falling off in numbers at 
great distances in high galactic latitudes as compared with 
low. The thinning out begins to be perceptible beyond 
100 parsecs, and at 250 parsecs the density is only about 
one-quarter of its value in the lower latitude. The 
decrease is more rapid than would be expected from 
Seeliger's figures. The results for Type A are not so easy 
to accept. I am inclined to believe, however, that there 
must be a real clustering of Type A stars in the low region 
at a distance of about 100 parsecs, as the figures 
suggest. Perhaps the cluster has special properties as 
regards motion, which will upset the details of our table. 
In the region of high latitude (for which there seems no 
good reason to doubt the results) the stars appear much more 
spread out than Type K, as would be expected if Type 
A is rarer in space but of greater average luminosity. It 
seems possible that the unexpectedly low average distance 
of Type A (see p. 168) may be due to the presence of 
extensive clusters of these stars near the Sun. It was 
remarked that the next preceding type, B, shows a 
great tendency to form moving clusters, and possibly 
the same phenomenon in a vaguer form may remain in 
Type A. 

For the distribution of the fainter stars the proper 
motions of 3,735 stars within 9 of the North Pole con- 
tained in Carrington's Circumpolar Catalogue are available. 
The limiting magnitude, which is the same as that of the 
Bonn Durchmusteruug, is about 10 m '3. These stars have 
been investigated by F. W. Dyson by a method similar 
in principle to that we have just described. Dyson used 
two forms of / (>), viz. 

(a) /(r) = re-* 8 ** 
(6) /(r) = 7-0-s e - 



226 



STELLAR MOVEMENTS 



CHAP. 



The differences in the resulting distributions are small ; 
the former probably gives the closer representation of the 
nearer stars, and the latter of the more distant stars. 
Both results are given in Table 38. 



TABLE 38. 
Distribution of Stars brighter than 10 m> 3 (for Galactic Latitude 27). 



Limits of Distance. 


Percentage of Stars. 


Parsecs. 


Formula (a). 


Formula (6). 


0- 40 


0-9 


1-0 


40- 100 


5-0 


4-7 


100 200 


15'1 


13-2 


200 400 


40'1 


35-2 


400 667 


31-5 


32-9 


667-1000 


7-1 


11-9 


>1000 


0-3 


1-1 



As might be expected, these stars are considerably more 
distant than those of Boss's Catalogue (cf. Table 37). 
For example, only 57 per cent, are within 100 parsecs of 
the Sun, as compared with more than 40 per cent, of the 
Boss stars. 

It may also be noted that 70 per cent, of the whole 
number have parallaxes between 0"*005 and 0"*0015 ; or, in 
terms of distance, 70 per cent are distant between 200 
and 650 parsecs. 

It is believed that in low galactic latitudes the actual 
density of the stars in space is fairly constant up to a very 
considerable distance from the Sun. On this hypothesis 
Tables 37 and 38 enable us to determine the luminosity 
law ; for the limiting magnitude of the stars considered 
corresponds to a limiting absolute luminosity which 
decreases as the square of the distance. If in the foregoing 
investigations 



STATISTICAL INVESTIGATIONS 



227 



/() dr is the number of stars between the limits of distance r to r+dr 

in an area of the sky o>, 

in is the limiting magnitude of the catalogue, 
Q^'o is the Sun's stellar magnitude if removed to a distance of 1 parsec, 

then the limiting luminosity for stars at a distance r is 

(2 -512) 0-5 -m x r-', 

and the number of stars per cubic parsec with luminosities 
greater than this is 

&). 

or/ - ~ 

In Table 39 are given the luminosity laws derived from 
the Boss proper motions (Type K only) and the Carrington 
proper motions (all types). Owing to the difference in 
brightness of the stars used, the two investigations apply 
mainly to different parts of the luminosity curve. 



TABLE 39. 

Number of Stars in 4'2 x 10 6 units of space equal to a sphere of radius 

100 parsecs. 



Boss Stars (Type K only). Carrington Stars (All Types). 


Luminosity Number Luminosity 
(8un=l). of Stars. (Sun = l). 


Number 
of Stars. 


>500 10-8 


>200 


400 to 500 7-3 | 100 to 200 


24 


300 400 12-9 


50 100 


316 


200 300 26-3 


25 50 


1,190 


150 200 25-0 


10 25 


3,310 


100 150 


51-9 


1 10 


18,360 


50 100 


139-0 


o-i i 


70,100 


25 50 


251-9 






10 25 


500 






1 10 


2,700 






Total brighter than 




Total brighter than 




the Sun .... 


3,725 


the Sun .... 


23,200 



The agreement is not particularly good. We should 
expect the numbers for Type K only to be considerably 

Q 2 



228 STELLAR MOVEMENTS CHAP. 

less than for all the types taken together, so that for 
luminosities less than 100 there is no noteworthy discord- 
ance. For the Carrington stars, the number with 
luminosity greater than 100 depends on stars distant 
more than 900 parsecs ; as these are not more than 2 per 
cent, of the whole, it is doubtful if the formulae ought to 
be pressed so far. Also it may well be that at so great a 
distance there is a real diminution of the density of dis- 
tribution of the stars in space. 

It is interesting to note that more than 95 per cent, of the 
stars of Carrington (or of the Bonn Durchmusterung), and 
nearly 99 per cent, of the stars brighter than 5 m *8, are 
more luminous than the Sun. For a star of brightness 
equal to the Sun to appear as bright as 10 m< 3 or 5 m> 8 it 
must be distant less than 91 or 11 parsecs, respectively. 
The percentages given are the proportions outside these 
limits. 

Adopting the law of distance-distribution for stars down 
to a limiting magnitude 



we may investigate the velocity-law in the direction of 
star-streaming. This has been done by Dyson for the 
Carrington proper motions. He found the best general 
agreement was given by Schwarzschild's ellipsoidal hypo- 
thesis rather than by the two-drift hypothesis. The ratio 
of the two axes of the velocity ellipse was found to be 
0*60, in close agreement with the investigations of much 
brighter stars by the methods of Chapter VII. The two- 
drift hypothesis does not represent the small proper 
motions at all closely, though the existence of two maxima 
in the observed distribution of motions lends support 
to the view that there is some sort of separation of 
the two streams. The disagreement of the small proper 
motions is perhaps not surprising, for it is just this defect 
'hat the third Drift was introduced to remedy ; but it is 
certainly to the credit of the ellipsoidal hypothesis that 



x STATISTICAL INVESTIGATIONS 229 

with so few constants it can lead to a good agreement with 
the small and large motions simultaneously. 

The main points of discrimination between the two- 
drift and ellipsoidal hypotheses appear to be 

(1) The skewness of the velocity-distribution. 

(2) The spread of the distribution. 

(3) The existence (or not) of two maxima. 

In the first point the advantage is admittedly with the 
two-drift hypothesis, and it seems equally certain that the 
ellipsoidal hypothesis is better fitted to the second. With 
regard to (3) the evidence is rather in favour of the two 
drifts. According to the relative importance attached to 
these three criteria, different views of the merits of the 
two methods of approximation will prevail. 

One point of great practical importance must be at- 
tended to in investigations of the kind we have been 
considering (Section III.). The distribution of the proper 
motions is generally appreciably altered by the presence 
of accidental error in the observations. Usually the pro- 
bable error of the observations is known, and in this case 
we can, from a table of the number of observed proper 
motions between given limits, form a revised table giving 
the true number, i.e., we can correct the observed values 
of h(rf) for the effects of the known accidental error. 6 
Unless the accidental error is large, the correction to be 
applied to each number of the observed table is 

_/r046x probable errory x ^^ ^^ difference 
\ tabular interval / 



18- 



The full formula, applicable to any kind of statistics, 

-> -(-)** 

where v (m) and u (m) are the true and observed frequency 
functions, and 0'477/A is the probable error of the observa- 
tions of m.- 



230 STELLAR MOVEMENTS CHAP. 

GENERAL CONCLUSION. 

The investigations described in this chapter suffer 
from all the imperfections inseparable from pioneer 
work. The numerical results in Sections I. and II. 
would, I believe, be modified in an important degree, 
by a modern determination of the mean parallaxes of 
stars of different magnitudes ; those of Section III. 
suffer from scantiness of data, insufficient range, and the 
absence of an effective check on the main assumptions. 
Yet I do not think it can be doubted that these general 
statistical researches have already greatly advanced our 
knowledge of the distribution and luminosities of the 
stars. If the approximation is not yet a close one, our 
present vague knowledge is very different from the 
complete ignorance from which we started. But the main 
interest of this chapter lies in the hope for the future. 

It is of special importance to note that there exist two 
entirely independent methods of determining the distribu- 
tion in distance of stars brighter than a limiting magnitude, 
the one resting on magnitude-counts and mean parallactic 
motions (Section I.),* and the other on the distribu- 
tion of the individual proper motions (Section III.). 
There is nothing in common between the data used for 
these two methods ; and the one is a complete check 
on the other. When the time arrives that this check 
is satisfied, and that results obtained along one line of 
investigation are in full agreement with those obtained 
along an independent line, the results of these methods of 
research will have been placed on a firm basis. Meanwhile, 
the conclusion that such a check is possible may be 
regarded as one of the most useful results of these 
preliminary discussions. 

* The method of Section I. leads, as we have seen, to a complete solution 
for the luminosity-law and density-law, The distribution in distance of stars 
brighter than a limited magnitude is easily derived from these. 



x STATISTICAL INVESTIGATIONS 231 

REFERENCES. CHAPTER X. 

1. Charlier, Lund Meddelanden, Series 2, No. 8, p. 48. 

2. Eddington, Monthly Notices, Vol. 72, p. 384. 

3. Comstock, Astron. Jour., No. 655. 

4. Seeliger, Sitzungsberichte, K. Bayer. Akad. zu Munchen, 1912, 'p. 451. 

5. Kapteyn. Astron. Journ., No. 566, p. 119. 

6. Eddington, Monthly Notices, Vol. 73, p. 359. 

BIBLIOGRAPHY. 

SECTIONS I. AND II. Kapteyn's principal investigations are : 
Groningen Publications, No. 8 (1901) Mean parallaxes. 

,, ,, No. 11 (1902) Luminosity and density laws, 

with further developments and revision in 
Astron. Journ., No. 566 (1904). 

Proc. Amsterdam Acad. Sci., Vol. 10, p. 626 (1908). 

Seeliger's investigations of the luminosity and density laws are contained 
mainly in four papers. 

K. Bayer. Akad. der Wiss. in Munchen, Abhandlungen, Vol. 19, Pt. 3 
(1898), and Vol. 25, Pt. 3 (1909) ; ibid., Sitzungsberichte, 1911, p. 413, and 
1912, p. 451. 

The most important parts of the mathematical theory are given very 
concisely by 

Schwarzschild, Astr. Nach., Nos. 4422 and 4557. 

SECTION III. The subject is treated by 

Dyson, Monthly Notices, Vol. 73, pp. 334 and 402. 

Eddington, Monthly Notices, Vol. 72, p. 368, and Vol. 73, p. 346. 
Discussions by methods different in the main from those here described 
are given by 

Charlier, Lund Meddelanden, Series 2, Nos. 8 and 9. 

v. d. Pahlen, Astr. Nach., No. 4725. 



CHAPTER XI 

THE MILKY WAY, STAR-CLUSTERS, AXD NEBULAE 

AT the beginning of the preceding chapter we emphasised 
the fact that the statistical investigations referred to an 
idealised sidereal system, which retained some of the more 
important properties of the actual universe, but neglected 
many of the details of the distribution. If an impression 
has been given that the spheroidal distribution of stars 
with density diminishing outwards is a complete and 
sufficient model, a glance at Plate I (Frontispiece), 
which is a photograph of the region of the Milky Way in 
the neighbourhood of Sagittarius, may serve to correct 
this. In the Milky Way there are unmistakable signs of 
clustering and irregularities of density on a large scale. 
The great star-clouds and deep rifts are features in 
marked contrast with the phenomena of distribution 
that we have hitherto considered, and no elaboration of 
the theory of a disc- shaped or spheroidal system will suf- 
fice to explain them. This does not affect our conclusions 
as to the shape of what we have called the inner stellar 
system ; a general concentration of stars to the galactic 
plane is manifested quite apart from the great clusters 
of the Milky Way itself. It would certainly be desirable 
in discussing problems such as those of the last chapter to 
ignore, or at least treat separately, the parts of the sky 
through which the Milky Way itself passes, for our 
idealised system evidently becomes inadequate here. 



CH. xi CLUSTERS AND NEBULA 233 

The view then is taken that there is first an inner stellar 
system consisting of a flattened distribution of stars of 
density more or less uniform at the centre and diminishing 
outwards ; and secondly a mass of star-clouds, arranged 
round it and in its plane, which make up the Milky Way 
(see Fig. 1, p. 31). It is to the inner system that our 
knowledge of stellar motions and luminosity relates. 
Whether the outer clouds are continuous with the inner 
system or whether they are isolated, is a question at 
present without answer. 

It is usual to consider the system in the midst of which the 
Sun lies as the principal system, the clusters of the Milky 
Way being a kind of appanage. An alternative view makes 
no such distinction, but contemplates a number of star- 
clouds scattered irregularly in the one fundamental plane, 
our own system being one of them. There are certain ad- 
vantages in the latter view, especially as the two star-streams 
could then be accounted for by two of these star-clouds 
meeting and passing through one another. By a natural 
reaction from the geocentric views of the Middle Ages, we 
are averse to placing the earth at the hub of the stellar 
universe, even though that distinction is shared by 
thousands of other bodies. But it is doubtful if there is 
really any close resemblance between the Milky Way 
aggregations and that which surrounds the Sun. We do 
not recognise in them the oblate form flattened in the 
fundamental plane, which is so significant a feature of the 
solar star-cloud. They seem to be of a more irregular 
character and for that reason we prefer to adhere to a 
theory which regards them as subsidiary. 

The great mass of the stars shown in photographs of 
the Milky Way are very faint. We have no knowledge 
of their motions or their spectra, and even now there is 
but little accurate information as to their magnitudes and 
numbers. It is an important question whether some of 
the bright -stars, which are seen in the same region as these 



234 STELLAR MOVEMENTS CHAP. 

star-clouds, are actually in the clouds or only projected 
against them. The investigations of this point are rather 
contradictory ; but on the whole it seems probable that 
some stars of the sixth magnitude are actually situated in 
the Milky Way clusters. Certainly by the ninth magnitude 
we have begun to penetrate into the true galaxy, and the 
twelfth or thirteenth magnitude takes us right into the 
heart of the aggregations. Simon Newcomb l attacked the 
problem by comparing the density of the lucid stars, 
where the Milky Way background was respectively bright 
and faint ; he found that bright stars were most numerous 
where the background was bright. 

TABLE 40. 
Relation of Lucid Stars to the Milky Way (Newcomb). 





N. Hemisphere 
(Limiting Mag. 
6'3). 


S. Hemisphere 
(Limiting Mag. 
7-0). 


Mean star-density of whole hemisphere 
Star-density : darker galactic regions . 
,, brighter ,, ,, 


19-0 
20-4 
32-9 


327 
33-8 
79-4 



The star-density is given per 100 square degrees. The 
greater value in the southern hemisphere is due to the 
lower limit of magnitude in the catalogue which was 
used. 

The condensation of the stars to the brighter regions is 
very marked ; but some caution is needed in interpreting 
this result. There is no doubt that many of the dark 
patches in the Milky Way are due to the absorption of 
light by tracts of nebulous matter. To perform their 
work of absorption these tracts must lie on the nearer side 
of the Milky Way aggregations. Since Table 40 shows that 
the star-density in the darker regions is barely greater than 
the mean for the hemisphere, and therefore less than in a 
zone just outside the Milky Way, the dark matter must be at 



XI 



CLUSTERS AND NEBULA 



235 



least partly within the oblate inner system. Newcomb's 
result therefore teaches us that some of the sixth and seventh 
magnitude stars lie in and beyond the dark clouds ; but it is 
not conclusive that any of them lie in the bright aggrega- 
tions of the Milky Way. To prove the latter result we 
should have to show that the star-density in the bright 
regions is greater than could reasonably be attributed to 
the oblate shape of the inner system ; the figures suggest 
that this is so, but there is room for doubt. 

A similar discussion of fainter stars has been made by 
C. Easton, 2 who considered especially the part of the Milky 
Way in Cygnus and Aquila , where there is a wide range 
in the intensity of the light. A selection from his results 
is given in Table 41. 

TABLE 41. 
Relation of Stars to the Milky Way (Easton). 





Argelander Wolf 


W. Herschel 




Durchmusterumj Photographs 


Star-gauges 




(Mag. 010). 


(Mag. 011). 


(Mag. 014). 


Star-density 






Darkest patches 23 


72 


405 


Intermediate ,, 33 


134 


4114 


Brightest ,, 48 


217 


6920 



The density is given per square degree. 

The numbers show that as we proceed to fainter stars a 
rapidly increasing proportion is associated with the true 
Milky Way aggregations, and by the time the fourteenth 
magnitude is reached, an overwhelming proportion is 
found to belong to them. But Easton's results and 
Xewcomb's are not quite in agreement. The relative 
superiority of the bright patches found by Easton for 10 m 
is barely greater than that found by Newcomb for 6 m -7 m . 
Indeed by extrapolation of Easton's figures we should 
conclude that the stars brighter than 7 m were not notice- 



236 STELLAR MOVEMENTS CHAP. 

ably associated with the Milky Way background. If we 
could suppose that the Cygnus-Aquila region is more 
distant than the average, the difference between the two 
results would be explained ; but Easton has given reasons 
for believing that this region is nearer than the average. 

In all the counts discussed by Easton. the star-density 
of the bright patches is so notably superior to the density 
just outside the Milky Way that we must conclude that 
the excess is actually due to the star-clouds. This method 
of considering the problem is so straightforward that it 
seems impossible to doubt the conclusions. The results 
should be independent of systematic errors in the limits 
of the counts, for the bright and dark regions adjoin one 
another and are irregularly intermixed. If there is any 
tendency to make the counts less complete in the rich 
regions or where the background is bright, this only 
means that the difference is really more accentuated than 
is shown in the Table. 

If it is established that the Milky AVay aggregations 
include a fair number of stars that appear to us of the 
ninth magnitude, it is possible to form some conception 
of their distance. A star of luminosity 10,000 times 
that of the Sun would appear of the ninth magnitude at 
a distance of 5,000 parsecs. This may be taken as an 
upper limit for the distance of the nearer parts of the 
Milky Way. If, following Newcomb, we admit the 
presence of sixth magnitude stars in the aggregations, the 
limit is reduced to 1200 parsecs. In any case the 
so-called li holes " in the Milky Way (dark nebulse) would 
seem to be in some cases within this latter distance. 
There is no reason to believe that all parts of the galaxy 
are at the same distance, and certain appearances suggest 
that there may be two or more branches lying behind one 
another in some parts of the sky. The relative bright- 
ness of the different portions gives no clue to the distance ; 
for the apparent brightness (per unit angular area) of a 



;< 



xi CLUSTERS AND NEBULA 237 

cluster of stars is independent of its distance.^ The 
differences in intensity must, therefore, be due either to a 
greater depth in the line of sight or to a closer concen- 
tration of the stars. (If there is appreciable general 
absorption of light in space, this statement would have 
to be modified.) Further, as it is necessary to suppose 
that the whole structure is quite irregular, it would be 
very unsafe to assume that the angular width of the 
different portions gives any measure of the distance. 

It is well known that the large irregular nebulae are 
found principally in the Milky Way, differing in this 
respect from the great majority of the compact nebulae. 
In the "irregular " class we include not only the more intense 
patches, such as the Omega, Keyhole, and Trifid nebulae, 
etc., but the extended nebulous backgrounds, such as those 
photographed by E. E. Barnard in Taurus, Scorpius, and 
other constellations. Of the same nature are many of the 
dark spaces in the Milky Way, where the light of the stars 
behind is cut off by nebulous regions which give little or 
no visible light of their own. These dark spaces are 
usually connected with diffused visible nebulosity, which 
often surrounds and is condensed about one or more bright 
stars. An excellent example is found in the constellation 
Corona Austrina (Plate 2), where there is a dark area con- 
taining very few stars, edged in some parts with visible 
nebulosity, which condenses into brig cusps round the 
bright stars. No one examining < photographs can 
doubt that the darkness is caused by the nebula the edge 
of which is visible. Visual observers have asserted that 
the region has a leaden or slightly tinted appearance as 
though a cloud were covering part of the field. 3 Another 
curious region is found in Sagittarius near the cluster 
Messier 22, where great curved lanes are, as it were, 
.smudged out among the thickly scattered stars. The 

* Nearness might decrease the background illumination somewhat, for 
a greater amount of the light would appear in the form of distinct stars. 



2 4 o STELLAR MOVEMENTS CHAP. 

association. In the Greater Cloud (Plate 3) there are a large 
number of nebulous knots, which have generally been 
described as spiral (i.e., non-gaseous) nebulae. If this were 
really their nature it would form a remarkable distinction be- 
tween the clouds and the Milky Way, for the spiral nebulae 
avoid the latter. But according to A. R. Hinks the 
supposed nebulae of the Magellanic Clouds are unlike any- 
thing found elsewhere, and have no resemblance to the true 
spiral nebulae. Many of the principal nebulae in the Cloud 
are undoubtedly gaseous. 

By an ingenious argument E. Hertzsprung 7 has arrived 
at an estimate of the distance of the Lesser Magellanic 
Cloud, which is entitled to some confidence. It depends 
on the existence of a large number of variables of the 
S Cephei type in that Cloud. Now there is reason to believe 
that the absolute magnitude of a Cepheid variable of given 
period is a fairly definite quantity ; that, in fact, it can be 
predicted from the period with a mean uncertainty of 
only a quarter of a m'agnitude. This was shown by Miss 
Leavitt 8 who discussed the variables in the Lesser Cloud. 
Since these must be at nearly the same distance from the 
Sun, their apparent magnitudes will differ from their 
absolute magnitudes by a constant. She found that the 
magnitude and the logarithm of the period were connected 
by a linear relation, and that the average deviation of any 
individual from the general formula was m *27. Now 
the mean distance of the brighter Cepheid Variables can 
be calculated from the paral lactic motion in the usual way. 
It is then only necessary to multiply by the factor for 
the difference of magnitude between these and the Magel- 
lanic Variables and allow for the difference of period, if 
any, in order to obtain the distance of the latter. In this 
way the distance of the Lesser Magellanic Cloud is found 
to be 10,000 parsecs the greatest distance we have yet 
had occasion to mention. 

Passing from the large-scale aggregations of stars, we 



Plate 3. 




Franklin-Adams Chart. 



THE GREATER MAGELLANIC CLOUD. 



xi CLUSTERS AND NEBULA 241 

must refer briefly to the star-clusters proper. There 
seems no reason to doubt that these are of the same nature 
as the moving clusters discussed in Chapter IV. In par- 
ticular the Taurus-stream may be taken to be typical of 
the globular clusters, although it is not one of the richer 
specimens of the class. The distribution of these globular 
clusters in the sky is very remarkable ; they are to be found 
almost exclusively in one hemisphere of the sky, the pole of 
which is in the galactic plane in galactic longitude 300. 
This result (which is taken from A. R. Hinks's discussion 9 ) is 
clearly of great significance ; but it does not seem possible 
at present to attempt any explanation of it. 

The chief characteristics, from our point of view, of the 
planetary nebulae are their close condensation to the Milky 
Way and the large radial velocities of those that have 
been measured. Here again we are not able to do more 
than state the facts. I am not aware of any trustworthy 
measures of the proper motions of planetary nebulae, and 
their size and distance are consequently a matter of extreme 
uncertainty ; but their marked tendency to lie in the 
plane of the Milky Way shows that they must be placed 
somewhere within our own stellar system. 

In Plate 4 is shown the Whirlpool Nebula in Canes 
Venatici, a fine example of the spiral nebulas, which are 
among the most beautiful objects in the heavens. It is 
generally believed that the spirals predominate enormously 
over the other classes of nebulae ; and, as the whole number 
of nebulae bright enough to be photographed has been 
estimated by E. A. Fath at 160,000, they must form a 
very numerous class of objects. They are seen by us at 
all inclinations, some, like the Whirlpool Nebula, in full 
front view, whilst others are edge-on to us and appear as 
little more than a narrow line. An example of the latter 
kind is also illustrated in Plate IV. In all cases, where it is 
possible to discriminate the details, the spiral is seen to 
be double-branched, the two arms leaving the nucleus at 

R 



242 STELLAR MOVEMENTS CHAP. 

opposite points and coiling round in the same sense. 
From the researches of E. v. d. Pahlen 10 it appears that 
the standard form is a logarithmic spiral. The arms, how- 
ever, often present irregularities, and numerous knots and 
variations of brightness occur. Unlike the planetary and 
extended nebulae, the spectrum shows a strong continuous 
background ; bright lines and bands are believed to occur, 
at least in the Great Andromeda Nebula ; but they are of 
the character of those found in some of the early type 
stars, and are distinct from the emission lines of the 
gaseous nebulae. 

The distribution of spiral nebulae presents one quite 
unique feature : they actually shun the galactic regions 
and preponderate in the neighbourhood of the galactic 
poles. The north galactic pole seems to be a more 
favoured region than the south. This avoidance of the 
Milky Way is not absolute ; but it represents a very 
strong tendency.' 

In the days before the spectroscope had enabled us to 
discriminate between different kinds of nebulae, when all 
classes were looked upon as unresolved star-clusters, the 
opinion was widely held that these nebulae were " island 
universes," separated from our own stellar system by a 
vast empty space. It is now known that the irregular 
gaseous nebulae, such as that of Orion, are intimately 
related with the stars, and belong to our own system ; 
but the hypothesis has recently been revived so far 
as regards the spiral nebulae. Although the same term 
" nebula " is used to denote the three classes irregular, 
planetary, and spiral we must not be misled into suppos- 
ing that there is any close relation between these objects. 
All the evidence points to a wide distinction between 
them. We have no reason to believe that the arguments 
which convince us that the irregular arid planetary nebulae 
are within the stellar system apply to the spirals. 

It must be admitted that direct evidence is entirely 



xi CLUSTERS AND NEBULAE 243 

lacking as to whether these bodies are within or without 
the stellar system. Their distribution, so different from 
that of .all other objects, may be considered to show that 
they have no unity with the rest ; but there are other 
bodies, the stars of Type M for instance, which remain 
indifferent to galactic influence. Indeed, the mere fact 
that spiral nebulae shun the galaxy may indicate that they 
are influenced by it. The alternative view is that, lying 
altogether outside our system, those that happen to be in 
low galactic latitudes are blotted out by great tracts of 
absorbing matter similar to those which form the dark 
spaces of the Milky Way. 

If the spiral nebulae are within the stellar system, we 
have no notion what their nature may be. That hypothesis 
leads to a full stop. It is true that according to one 
theory the solar system was evolved from a spiral nebula, 
but the term is here used only by a remote analogy with 
such objects as those depicted in the Plate. The spirals to 
which we are referring are, at any rate, too vast to give birth 
to a solar system, nor could they arise from the disruptive 
approach of two stars ; we must at least credit them as 
capable of generating a star cluster. 

If, however, it is assumed that these nebulae are external 
to the stellar system, that they are in fact systems coequal 
with our own, we have at least an hypothesis which can be 
followed up, and may throw some light on the problems that 
have been before us. For this reason the " island universe " 
theory is much to be preferred as a working hypothesis ; 
and its consequences are so helpful as to suggest a dis- 
tinct probability of its truth. 

If each spiral nebula is a stellar system, it follows that 
our own system is a spiral nebula. The oblate inner 
system of stars may be identified with the nucleus of the 
nebula, and the star clouds of the Milky Way form its 
spiral arms. There is one nebula seen edgewise (Plate IV) 
which m'akes an excellent model of our system, for the 

R 2 



244 STELLAR MOVEMENTS CHAP. 

oblate shape of the central portion is well-shown. From 
the distribution of the Wolf-Ray et stars and Cepheid 
Variables, believed to belong to the more distant parts of 
the system, we infer that the outer whorls of our system 
lie closely confined to the galactic plane ; in the nebula 
these outer parts are seen in section as a narrow rectilinear 
streak. The photograph also shows a remarkable absorp- 
tion of the light of the oblate nucleus, where it is crossed 
by the spiral arms. We have seen that the Milky Way 
contains dark patches of absorbing matter, which would 
give exactly this effect. Moreover, quite apart from the 
present theory, a spiral form of the Milky Way has been 
advocated. Probably there is more than one way of repre- 
senting its structure by means of a double-armed spiral ; 
but as an example the discussion of C. Easton n may be 
taken, which renders a very detailed explanation of the 
appearance. His scheme disagrees with our hypothesis 
iu one respect, for he has placed the Sun well outside the 
central nucleus, which is situated according to his view in 
the rich galactic region of Cygnus. 

The two arms of the spiral have an interesting meaning 
for us in connection with stellar movements. The form 
of the arms a logarithmic spiral has not as yet given 
any clue to the dynamics of spiral nebulae. But though 
we do not understand the cause, we see that there is a 
widespread law compelling matter to flow in these forms. 

It is clear too that either matter is flowing into the 
nucleus from the spiral branches or it is flowing out from 
the nucleus into the branches. It does not at present con-, 
cern us in which direction the evolution is proceeding. In 
either case we have currents of matter in opposite direc- 
tions at the points where the arms merge in the central 
aggregation. These currents must continue through the 
centre, for, as will be shown in the next chapter, the 
stars do not interfere with one another's paths. Here 
then we have an explanation of the prevalence of motions 



xi CLUSTERS AND NEBULAE 245 

to and fro in a particular straight line ; it is the line from 
which the spiral branches start out. The two star- 
streams and the double-branched spirals arise from the 
same cause. 

REFERENCES. 

1. Newcomb, The Stars, p. 269. 

2. Easton, Proc. Amsterdam Acad. Sci., Vol. 8, No. 3 (1903); Astr. 
Nach., Nos. 3270, 3803. 

3. Knox Shaw, The Observatory, Vol. 37, p. 101. 

4. W. Herschel, Collected Papers, Vol. 1, p. 164. 

5. Hertzsprung, Astr. Nach., No. 4600. See a Iso Newcomb "Contribu- 
tions to Stellar Statistics, No. 1" (Carnegie List. Pub., No. 10.) 

6. Harv. Ann., Vol. 56, No. 6 ; Newcomb, The Stars, p. 256. 

7. Hertzsprung, Astr. Nach., No. 4692. 

8. Leavitt, Harvard Circular, No. 173. 

9. Hinks, Monthly Notices, Vol. 71, p. 697. 

10. v. d. Pahlen, Astr. Nach., No. 4503. 

11. Easton, Astrophysical Journal, Vol. 37, p. 105. 



- 



CHAPTER XII 

DYNAMICS OF THE STELLAR SYSTEM 

DURING the time that the stars have been under 
observation their motion has been sensibly rectilinear 
and uniform. A reservation must be made in the case of 
binary stars, where the components revolve around one 
another ; but from the present point of view pairs or 
multiple systems of this kind only count as single 
individuals. With this exception, we have no direct 
evidence that one star influences the motion of another ; 
yet we cannot doubt that, in the vast period of time 
during which the stellar universe has been developing, the 
forces of gravitation must have played a part in shaping 
the motions that now exist. It may not be premature to 
consider the dynamical aspects of some of the discoveries 
of recent years. 

The action of one star on another, even at the smallest 
normal stellar distance is exceedingly minute. The attrac- 
tion of the Sun on a Centauri imparts to that star in the 
course of a year a velocity of one centimetre per hour. At 
this rate it would take 380,000,000 years to communicate 
a velocity of one kilometre per second. The period is not 
so excessive, compared with what we believe to be the 
life of a star, as to entitle us to despise such a force. But 
the two stars will not remain neighbours for more than a 
small fraction of that time. Although a Centauri is at 
present approaching, a separation will soon begin ; in 



CH. xii DYNAMICAL THEORY 247 

150,000 years from now the distance will have doubled; 
and before the communicated velocity amounts to more 
than a fraction of a metre per second the star will have 
receded out of range of the Sun's attraction. 

The case is different when we consider the general 
attraction of the whole stellar system on its members. 
Not only is the magnitude of the force somewhat greater, 
but the time through which its effects accumulate is far 
longer than in the case of one star acting on a temporary 
neighbour. This general attraction is quite sufficient to 
produce important effects on the stellar movements. 

The field of force in which a star moves is due to 
a great number of point-centres the stars. The distribu- 
tion of the attracting matter is discontinuous. We there- 
fore divide the force into two parts, (1) the attraction 
in an ideal continuous medium having the same average 
density, and the same large-scale variations of density 
as the stellar system, and (2) the force due to the 
accidental arrangement of the stars in the immediate 
neighbourhood. The same distinction occurs in the 
ordinary theory of attractions for points inside the 
gravitating matter. AVe call the first part the general 
or central attraction of the system ; the term central is 
perhaps inaccurate for there is no true centre of attraction 
unless the system possesses spherical symmetry. The 
second part is of an accidental character and will act in 
different directions at different times ; but it is not on 
that account to be dismissed without consideration. 

The stars have often been compared to the molecules of 
a gas ; and it has been proposed to apply the theory of 
gases to the stellar system. 1 The essential feature in gas- 
dynamics is the prominent part taken by the collisions of 
the molecules. Now it is clear that collisions of the stars, 
if they occur at all, must be exceedingly rare ; and the 
effect would certainly not be the harmless rebound 
contemplated by the theory of gases. It is, however, 



248 STELLAR MOVEMENTS CHAP. 

well-known that the precise mode of interaction during the 
encounter is of little importance, and all that is required 
in the theory is an interchange of momentum taking place 
between two individuals in their line of centres. In this 
generalised sense encounters are continually taking place ; 
the passage of one star past another always involves some 
interchange of momentum. It remains to be examined 
whether this continuous transference can play the same 
part in stellar theory as the abrupt changes of momentum 
in the gas theory. 

In the long run the same effect will be brought about. 
The ultimate state of a system of gravitating stars will be 
the same as that of a gas. The ultimate law of velocities 
will be the same as in a mass of non-radiating monatomic 
gas under its own attraction ; and moreover there will be 
equipartition of energy between the stars of different 
masses, just as if they were atoms of different weights. 
We might even go further , and look forward to a still 
more " ultimate " state, in which the double stars behaved 
as diatomic molecules. But it is unnecessary to pursue 
these deductions, for they have no reference to anything 
in the present state of the stellar universe, or to any 
future near enough to interest us. 

It was seen in Chapter IV. that the existence of the 
Moving Clusters shows plainly that the encounters have 
not as yet had any appreciable effect on the motions of the 
stars. Taking, for example, the Taurus Cluster, we have 
seen that it occupies a sphere of about 5 parsecs radius, 
which would in an ordinary way contain 30 stars. As it can- 
not be supposed that a special track has been cleared for 
the passage of the cluster, the stars th;it would naturally 
occupy the space must be there, permeating the cluster 
without belonging to it. hi so far as they have any 
effect at all, the attractions of these interlopers must tend 
to break up and dissipate the cluster, by destroying the 
parallelism of the motions. As no such breaking up has 



XII 



DYNAMICAL THEORY 



249 



taken place, it may be inferred that the chance encounters 
have had no appreciable effect on the stellar velocities up 
to the present time. Many of the stars of the Taurus 
Cluster are in a mature stage of development, so that this 
inference may fairly be applied to the general mass of 
the stars. 

A consideration of this question from the theoretical side 
is in entire accordance with this conclusion. We begin by 
considering the numerical amount of the deflection pro- 
duced by an encounter in given circumstances. 




Fig. 22. 

Let S l S. 2 (Fig. 22) be two stars of masses m l and m 2 and 
G be their common centre of gravity. 
Since 



we may replace the star S. 2 , in considering its attraction on 
S lt by a star of mass 

ml m * Yat. 

"A?^ + JI1 2 / 

Taking G to be at rest, let the star S L move along the 
hyperbolical path HAH f starting with an initial velocity 
V. Let CH, CH f be the asymptotes of the hyperbola. 

Draw G Y perpendicular to CH ; then G Y is equal to 
the transverse axis b of the hyperbola. 



2 5 o STELLAR MOVEMENTS CHAP 

The usual equation h 2 = pi gives in this case 



Hence 

a = -^s 
The deflection i|r = 

180- HCH' 
is given by 

or, since ty is always a small angle, 

, 2^ 
6F 2 ' 

The transverse velocity imparted by the encounter is 



If U is the initial relative velocity of the two stars, 
(T the distance of nearest approach calculated as if no 
deflection were to take place, 

F = 2 _ u, 

w*! + w 2 
h W 2 

m l + m 2 

Also 



m- \ 2 
? ) , 

l + m 2 / 



where 7 is the constant of gravitation. 
Hence the transverse velocity imparted is 



It may be noted that this expression does not involve 
m lt so that the tendency towards equipartition of energy 
is not indicated in the formula. Equipartition appears to 
be a third-order effect depending on >|r 3 , which has been 
neglected in the foregoing analysis. 

Close approaches which produce an appreciable deflection 



xii DYNAMICAL THEORY 251 

-fy- are exceedingly rare. Evidently the probability of such 
an event happening can be calculated when the density of 
stellar distribution is known. It is of greater importance 
to determine what is the cumulative effect of the large 
number of infinitesimal encounters experienced by a star 
in the course of a long period of time. The following 
discussion is based on an investigation by J. H. Jeans. 2 

We may set two limits, CT O and ov The former is the 
upper limit of distance for sharp encounters producing 
considerable deflections ; these will be treated as excep- 
tional events to be studied separately. The latter is an 
arbitrary limit beyond which approaches will not be held 
to constitute an encounter. 

Let v be the number of stars per unit volume. 

The mean free-path (as defined by Maxwell) is 

i 

V27T./0-! 2 ' 

Thus in a length of path L the total number of en- 
counters to be expected is 

N = 2*m,<r*L. 

The transverse velocity imparted at any encounter has 
been shown to be 

2ym. 2 
U<r' 

The average value of the relative velocity U is some- 
what greater than the velocity v of the star S l relative to 
the stellar system, for collisions will most frequently 
occur with those stars which are coming to meet S lf 

The, average value of I/a- 

= I - 2n<r da- -f- I 2rro- d<r 

J o " J o 

2/<r r 

As each encounter takes place in a haphazard direction, 
the individual contributions of transverse velocity must 
be compounded according to the theory of errors. Thus 
the probable resultant of N encounters is proportional 



252 STELLAR MOVEMENTS CHAP. 



to ^/N ; and, generally, the square of the probable re- 
sultant will be the sum of the squares of the indivi- 
dual deflections. Thus in averaging different sorts of 
encounters, we ought to use the mean- square values. 

The mean-square value of is somewhat greater than 

cr 

2 

.* We may conveniently regard this excess as roughly 
a i 

cancelling the excess of U over v and write the resultant 
transverse velocity after N encounters 



and the resultant deflection (in radians) 



Substituting the value of N, the deflection becomes 



The following numerical results are deduced by Jeans, 
on the assumption that the density of stellar distribution 
is one star to a sphere of radius one parsec, and that the 
average mass of a star is five times that of the solar 

* As the mean-square value appears at tirst sight to be infinite some 
further explanation may be desired. The ratio of the mean-square value of 

- to the simple mean is 

<T 



which is analytically infinite. 

If, however, we reserve sharp encounters for separate consideration wo 
may set <r () instead of for the lower limit. For a deflection of 2, <r will be 
about 500 astronomical units, o-j need certainly not be taken greater than 
500,000,000 astronomical units, for an i ncounter at that distance would last 
an indefinite time. With tln-M- values 



Thus the mean-s<|uarc \alue is nut more than 1M'> times the simple mean, 
sudden deflections of over '1 In-ill^ reserved. 



xii DYNAMICAL THEORY 253 

system. This density is somewhat greater than that which 
we have regarded as probable, and, accordingly, the results 
may exaggerate a little the disturbance due to encounters. 

Jeans finds that for an average star a deflection of 1 
may be expected after 3,200 million years In addition to 
this there is a small " expectation " of deflection by violent 
encounters. From such a cause a deflection of 2 or more 
might be expected once in a period of 8 x 10 11 years. The 
meaning of these figures may be illustrated by a definite 
instance. Considering a moving cluster, all the stars of 
which have equal and parallel velocities of 40 km. per sec., 
let a star be considered to continue a member of the main 
stream so long as its direction of motion does not diverge 
by more than 2. After 100 million years only 1 in 8,000 
of the original members will be lost by violent encounters, 
and the remainder will make angles with the main stream 
of which the average amount is only 10'. After 3,200 million 
years the loss will be 1 in 250, and the average angle of the 
remainder will be 1. After 80,000 million years one-tenth 
of the original members have been lost by sharp encounters, 
but the average angle of the remainder is 5. The ultimate 
dissolution thus takes place mainly by gradual scattering 
and not by sharp encounters. 

It must riot be overlooked that a cluster possesses a 
certain cohesion of its own, which may resist the minute 
scattering force to which it is subjected. The cluster is a 
place where the stars are grouped more densely than in 
the rest of space and a gravitational force is exerted on 
those which tend to stray away from it. As we cannot 
even roughly estimate the whole number of stars in any of 
the clusters, it is not possible to determine the amount of 
this force ; but a simple calculation will show that it must 
play some part in keeping the cluster together. Taking 
Boss's Taurus-stream, which moves at the rate of 40 km. 
per second, it has been stated that a deflection of 1 may 
be expected after 3,200 million years, which is equivalent 



STELLAR MOVEMENTS CHAP. 

to 1' after a million years. This deflection is equivalent 
to a transverse velocity of 0*012 parsec per million years. 
It is easily shown that the probable transverse displace- 



f . Ar - ir . n i 0-012 x 

ment after Jy million years will be - - - - parsecs. 



At the present time the stars of this cluster have spread 
away from the mean position by an average distance of 
about 3 parsecs. The corresponding value of 2V is 57. 
This calculation gives 57 million years as an upper limit 
of the age of the Taurus Cluster, assuming that its present 
extension is wholly due to encounters. The result 
depends on the rather high value of the stellar density 
used by Jeans, but with a much smaller density the period 
is still unreasonably short. It is clear therefore that the 
dissolving effect of the encounters has been very largely 
counteracted, and presumably the opposing circumstance is 
the mutual gravitation of the members of the cluster. 

This consideration does not destroy the force of our 
previous argument. The cohesion of the cluster is only 
important because the dissolving forces are so excessively 
minute. Jeans's calculation applies directly to an inde- 
pendent star, and shows that it can pursue its course 
practically unmolested ; whilst the observational evidence 
is that the effect of encounters is so small that even the 
minute attraction in a moving cluster is sufficient to 
counteract it. The evidence, observational and theoretical, 
seems so conclusive that we have no hesitation in accepting 
it as the basis of stellar dynamics. The apparent analogy 
with the kinetic theory of gases is rejected altogether, and 
it is taken as a fundamental principle that the stars 
describe paths under the general attraction of the stellar 
system without interfering with one another. 

Let us now attempt to estimate the order of magnitude 
of the general attraction, or the time required for a star 
to describe its orbit. If the stellar system is not spheri- 
cally symmetrical the orbits will not in general be closed 



xii DYNAMICAL THEORY 255 

paths. But for our estimates we need no precise definition 
of the periodic time ; we want to know roughly how long 
it takes for a star to pass from one side to the other of the 
sidereal system and back again. Within a sphere of 
uniform density all stars would describe elliptic orbits 
about the centre isochronously, whatever the initial con- 
ditions. The period depends only on the density and 
is independent of the size of the sphere. The greater 
the density the less will be the period according to the 
relation T oc p~*. In an ellipsoidal system, the component 
motions along the principal axes will be simple harmonic 
but with differing periods ; the period in a spherical 
system of the same density will be intermediate between 
them. Thus the period calculated simply from the density 
on the assumption of a spherical distribution will give 
a general idea of the period in the actual universe. 

\Yc have estimated the number of stars in a sphere of 
radius 5 parsecs to be 30 ; as these stars are mostly fainter 
than the Sun, we shall take the mass in this sphere to be 
only 10 times the Sun's. If this is an underestimate, and 
it may be very much underestimated on account of the 
possible presence of dark stars, the period obtained will 
be an upper limit. With the adopted density the result 
is 300,000,000 years. This is less than current estimates 
of the age of the solid crust of the earth. Thus the Sun 
and other stars of like maturity must have described at 
least one and probably many circuits, since they came 
into being. * We are justified in thinking of the stellar 
orbits as paths that have actually been traversed, and 
not as mere theoretical curves. 

The problems on which dynamics would be expected 
to throw some light are numerous. Why have the stars 

* We have no means of estimating the age of the stellar system, but 
perhaps there is no harm in having some such figure as 10 10 years at the 
back of our minds in thinking of these questions. 



256 STELLAR MOVEMENTS CHAP. 

in the early stages very small velocities ? Why do these 
velocities afterwards increase ? In particular, how do the 
stars acquire the velocities at right angles to the original 
plane of distribution, which cause the latest types to be 
distributed in a nearly spherical form ? How are the two 
star-streams to be explained ? What is the meaning of 
the third stream, Drift ? Can the partial conforming 
to Maxwell's law be accounted for ? What prevents the 
collapse of the Milky Way ? 

Some of these problems seem to be at present quite 
insoluble. Indeed, it must be admitted that very little 
progress has been made in the application of dynamics 
to stellar problems. What has been accomplished is 
rather of the nature of preparatory work. It has been 
shown that stellar dynamics is a different study from gas- 
dynamics, and, indeed, from the theory of any type of 
system that has yet been investigated. A regular pro- 
gression may be traced through rigid dynamics, hydro- 
dynamics, gas-dynamics to stellar dynamics. In the first 
all the particles move in a connected manner; in the 
second there is continuity between the motions of con- 
tiguous particles ; in the third the adjacent particles act 
on one another by collision, so that, although there is no 
mathematical continuity, a kind of physical continuity 
remains ; in the last the adjacent particles are entirely 
independent. A new type of dynamical system has 
therefore to be considered, and it is probably necessary 
first to work out the results in simple cases and to become 
familiar with the general properties, before attempting to 
solve the complex problems which the actual stellar 
universe presents. This has been the mode of develop- 
ment in the other branches of dynamics. 

The natural starting point is to investigate the possible 
steady states of motion. It will be understood that we 
are not here referring to the ultimate steady state in which 
the gas-distribution of velocities prevails, but a state which 



xii DYNAMICAL THEORY 257 

remains steady so long as the effect of the encounters is 
negligible. The actual stellar system may or may not be 
in such a state ; we can best hope to settle the question by 
working out the consequences of making that hypothesis. 

For systems possessing globular symmetry a number of 
types of steady motion have been found and investigated. 3 
None of those discovered up to the present give a reasonable 
approximation to the actual distribution of velocities ; but 
the failures seem to narrow down very considerably the 
field in which possible solutions may be obtained. A self- 
consistent dynamical model, possessing some at least of 
the main features of stellar motions, would be a most use- 
ful and suggestive adjunct in many kinds of investigation, 
and it would seem to be well worth while continuing the 
search for suitable systems. 

There is one problem relating to the actual stellar 
system which may be kept in view even at this early 
stage. H. H. Turner 4 has made an interesting suggestion, 
which gives a possible explanation of the two star-streams. 
The problem is to account for preferential motion to and 
fro in one particular line. Suppose that the stars move in 
orbits which are in general very elongated, somewhat like 
the cometary orbits in the solar system. The stellar 
motions then will be preferentially radial rather than 
transverse. If the Sun is at a considerable distance from 
the centre of the stellar system, the line joining that 
centre to the Sun will be a direction of preferential motion 
for those stars which are sufficiently near us to have 
sensible proper motion. Even if the eccentricity of the 
Sun is not very great, an effect of the character of star- 
streaming will be observed. We have always assumed 
that the convergence of the apparent directions of star- 
streaming in different parts of the sky was evidence that the 
true directions were parallel ; but a convergence of the 
true directions is an equally possible interpretation. It is 
quite possible that the preferential motions may be towards 



258 STELLAR MOVEMENTS CHAP. 

or away from a point at a finite distance rather than 
parallel to a line. It is difficult to say whether such a 
hypothesis would prove satisfactory in detail ; but at least 
there is no obvious objection to it. 

It may be asked, Why should the stellar orbits be very 
elongated ? The reason that may be assigned is that they 
start initially with very small velocities. We must sup- 
pose that their velocities in later stages are mainly acquired 
by falling towards the centre of the system, and it is only 
natural that they should be preferentially radial. Indeed, 
surprise has sometimes been expressed that there is so little 
indication of the preponderating radial movements that 
might be expected. Is it possible that the perplexing 
phenomenon of the two star-streams is just this indication? 

The great difficulty is that, if the motions are mainly 
radial, it seems inevitable that there should be a great 
congestion of stars in the neighbourhood of the centre- 
greater than we care to accept as possible. In the systems 
that have been worked out up to the present, it has nob 
been possible to obtain sufficient preferential motion with- 
out too great a density at the centre ; but we cannot as yet 
conclude that this holds generally. Of course, it is possible 
to argue that one of the dense patches of stars either in the 
Cygnus or Sagittarius regions of the Milky Way is the 
actually congested centre of the stellar system. 

A few remarks may be made on the other problems 
suggested on p. 256. The birth of a star without motion 
does not seem to present so much difficulty as has some- 
times been supposed. It is not necessary to suppose that 
the primordial matter from which it arises is not subject to 
gravitation (though there is nothing inherently improbable 
in such a speculation). Presumably a star is formed by the 
gathering together of meteoric or gaseous material in some 
portion of space. Now, we know that, if we were to lump 
together a thousand stars, their individual motions would 



xii DYNAMICAL THEORY 259 

practically cancel and the resultant super-star would be 
nearly at rest. Similarly, in forming a single star, the 
individual motions produced by gravitation in the materials 
of which it is composed might cancel so that the star would 
start from rest. It is interesting to note that by this 
process the average initial velocity of a star composed of 
A" constituents would vary ceteris paribus as JV~*, the 
velocities of the constituents being assumed to be 
haphazard. This might appear to lead to an equiparti- 
tion of energy between stars of different masses, the mass 
being proportional to N. But unless the number of 
constituents is very small their velocities would have to be 
enormous, and the suggestion does not seem to be tenable. 
Moreover, it is difficult to see how a moving cluster could 
be formed.* 

An increase of velocity in. the next stage is the natural 
re>ult of the central attraction of the stellar system. If 
the star starts existence with no motion, it must start from 
apcentron, and at all other points of the orbit its velocity 
would be greater. After the star is old enough to have 
described one quadrant of its orbit, we cannot look for 
any increase in velocity from this cause ; in other words, 
the progressive increase of velocity should cease after the 
first 100,000,000 years. But we can scarcely compress 
the development from Type B to Type M into that period. 
Another difficulty is that the motion produced by the 
central attraction would be mainly in the plane of the 
galaxy ; there is no explanation of the motions perpendi- 
cular to that plane acquired by the stars of the later types. 
It does not seem possible to account for these extra- 
galactic motions in any simple way. 

I see no alternative to supposing that the K and M 
stars have been formed originally in a more globular 
distribution than the early type stars. It may be that the 

* This suggestion (with the objections to it) was mentioned to me by 
Prof. A. Schoster. 



260 STELLAR MOVEMENTS CHAP. 

birth of stars was for some reason retarded in the galactic 
plane, and that this is the reason why early type stars 
abound there. Perhaps a more likely supposition is that 
massive stars with slow development have formed where 
the material was rich, and small stars with rapid develop- 
ment have formed where the star-stuff was scanty. Thus 
the outlying parts of the stellar system away from the 
galaxy have given birth to the small stars which have 
rapidly reached the M stage ; and these, having fallen in 
from a great distance, have acquired large velocities. The 
regions of the galactic plane, richly supplied with the 
necessary material, have formed large stars, which are 
only slowly developing, and these have remained moving 
in the galactic plane. The hypothesis in outline seems 
fairly plausible ; but so long as the difficulty of the double 
character of the Type M stars remains we cannot regard 
any explanation as complete. 

The foregoing suggestion is also applicable if we adopt 
Russell's hypothesis (p. 170). His view is that only the 
most massive stars are able to heat themselves up to the 
high temperatures characteristic of Types B and A, 
Accordingly, these types will only have originated 
where the star-forming material was rich, and their 
galactic concentration and small velocities can be 
accounted for. 

The problem of the equilibrium of the Milky Way is 
another subject for reflection. It seems necessary to grant 
that it is in some sort of equilibrium ; that is to say, the 
individual stars do not oscillate to and fro across the stellar 
system in a 300,000,000 year period, but remain concen- 
trated in the clusters, which they now form. The only 
possible explanation in terms of known forces is that the 
Milky Way as a whole is in slow rotation, a condition 
which has been considered by H. PoincareY 5 To obtain 
some notion of the order of magnitude of the rotation, let 
us assume the total mass of the inner stellar system to be 



xii DYNAMICAL THEORY 261 

10 9 times the Sun's mass, and the distance of the Milky 
Way to be 2,000 parsecs ; the angular velocity for equili- 
brium will then be 0"'5 per century. It may be pointed 
out that Charlier 5 has found that the node of the invariable 
plane of the solar system on the plane of the Milky Way 
has a direct motion amounting to 0"'35 per century, a 
motion which might equally well be expressed as a rotation 
of the stars in the plane of the Milky Way in a retrograde 
direction. Perhaps it would be straining the result 
too far to regard this as evidencing the truth of our 
surmise. 

With this brief reference to the dynamical aspect of the 
problem, we conclude our survey of the structure of the 
stellar system. The results discussed have, with few 
exceptions, come to light during the last fifteen years ; 
but they are the outcome of a century's labour of 
preparation. The proper motions now used are based on 
observations which go back to the time of Bradley ; and 
the modern instrumental methods, which are now yielding 
parallaxes and proper motions for discussion, have a long 
history of gradual development behind them. The 
progress of stellar investigation must not be measured by 
the few conclusions to which we have been able to give 
definite statement. In the future the fruits of these 
labours will be reached far more fully. 

Meanwhile the knowledge that has been attained shows 
only the more plainly how much there is to learn. The 
perplexities of to-day foreshadow the discoveries of the 
future. If we have still to leave the stellar universe a 
region of hidden mystery, yet it seems as though, in our 
exploration, we have been able to glimpse the outline of 
some vast combination which unites even the farthest 
stars into an organised system. 



262 STELLAR MOVEMENTS CHAP, xn 

REFERENCES. CHAPTER XII. 

1. Cf. Poincare, 7/<//<//,;>r.s Cosmogcniques, p. '257. 

2. Jeans, Monthly Notices, Vol. 74, p. 109. 

3. Eddington, Monthly Notices, Vol. 74, p. 5. 

4. Turner, Monthly Notices, Vol. 72, pp. 387, 474. 

5. Charlier, Lund Meddelanden, Series 2, No. 9, p. 78. 

6. Poincare, Hypotheses Cosmogoniques, p. 263. 



INDEX 



ABSORBING matter, 237 

Absorption of light in space, 33, 203, 

217 

Adams, W. S., 63, 155 
Age of moving clusters, 254 
Aity's method (solar motion), 79 
Aitken, K. G., 178 
Antares, 45, 49 
Apex, Solar 

position of. 73, li>2 

theory of determining, 79 
Apices of star-streams 100, 123 

summary of determinations, 125 
Arcturus, 45, 51 
Auwers, G. F. J. A., 207 

B TYPE stars, 160, 179 

Backhouse, T. W., 65 

Barnard, E. E., 237 

Beljawsky, S., 124 

Bellamy, 'F A., 19 

Bessel's method (solar motion), 83, 

119 

Betelgeuse, 49, 63 
Bibliographies, 28, 70, 85, 153, 231 
Binary stars, densities, 23, 174 

evolution, 178 

masses, 22, 160 

number of, 44 

relation between period and tvpe, 
178 

physical connection, 54 
Bohi-; N., 8 
Boss, B., 64 
Boss, L., masses of binaries, 23 

solar apex. 73, H>2 

spectral types, 158, 167, 180 

star-streams, 124 

Taurus cluster, 56 

Boss's 'Preliminary General Catalogue,' 
18, 93 

C CHAKACTKRISTK , 175 

Campbell, W. W., distances of types, 

167 

Maxwellian law, 152 
periods of binaries, 178 



Campbell, W. W., solar motion, 73, 162 

spectroscopic binaries, 44 

star-streams, 107 

velocities of Types, 156 
Canopus, 45 

a Centauri, 15, 19, 21, 22, 41, 50, 246 
Cepheid variables, 6, 240 
Chapman, S., 5, 186 
Charlier, C. V. L., 124, 207, 261 
Clusters, 241 
Clusters, Moving, 54 
Colour index, 11 

Comstock, G. C., 19, 76, 124, 207 
Convergence of stream motions, 99 
Corona Austrina, 237 
Counts of stars, 184 
Cross proper motions, 158, 166 
61 Cygni cluster, 70 
Cvgnus region of Milky Way 235. 244, 

258 
C.Z. 5 h 24318, 20, 41, 50 

DARK nebulas, 237 
Densities of stars, 23, 174 
Density of stellar distribution near the 
Sun, 44 

density law, 201, 214, 217 
Distances of bright stars, 224 

Carrington stars, 226 

Magellanic Cloud, 240 

Milky Way. 236 

near stars/41, 47 

spectral types, 167 

two star-streams, 110, 139 
Distribution of nebulae, 27, 241, 242 
Distribution of stars in distance, 222 

relative to Galactic plane, 165, 198 
Donner, A. S., 60 
Doppler's principle, 19 
Double stars. See Binary stars 
Draper classification, 7 
Drift, definition of, 89 
Drift O, 121 
'Dwarf" stars, 170 
Dynamics, stellar, 25! i 
Dvson, F. W. , Carrington stars, 19, 
225, 228 






264 



INDEX 



Dyson, F. W., Groombridge stars, 75 
'Stars of large proper motion, 42, 104, 
124, 155 

EASTON, C., 235, 244 
Effective wave-lengths, 12 
Ellipsoidal hypothesis, 103, 130 

comparison with two-drift hypo- 
thesis, 133, 229 

radial velocities, 145 
Encounters of stars, 249 
Equilibrium of Milky Way, 260 
Equipartition of energy, 159, 247 
Error-law for luminosities, 173 

parallaxes, 206 

velocities, 127, 151 

Errors of observation, effect of, 147, 229 
Espin, T. E., 181 
Evolution of binary systems, 178 

spectral types, 7, 176 

stellar system, 259 

FATH, E. A., 27, 241 
Fourier integrals, 209 
Fourth type stars, 181 
Fowler, A , 8 
Franklin- Adams, J., 191 
Frost, E. B., 44, 155 
Functions, tables of /(T), 129 

!), 132 
141 



GALACTIC longitude, variations of star 

density, 196 
Galactic plane, 31 

concentration of stars, 191 

concentration of special objects, 165, 
239 

relation of drifts to, 117 
Galactic pole, position of, 239 
Gas theory, 159, 247 
"Giant" and "Dwarf" stars, 170 
Gill, SirD., 181, 191 
Gravitation, interstellar, 246 
Groombridge 34 comes, 6 
a Gruis, 181 

HALM, J., constant of precession, 150 
equipartition of energy, 159 
two-drift theory, 117, 124 
three-drift theory, 118, 121 

Harvard Standard Sequence, 5 

Helium stars, see B type stars 

Herschel, Sir J., 192 

Herschel, Sir W., 73, 238 

Herschel's star-gauges, 191, 192, 213 

prung, E.. '--characteristic, 175 
effective wave-lengths, 12 
"giant" and "dwarf" stars, 170 
Lesser Magellanic Cloud, -Jin 
objects with galactic concentration, 

239 
Ursa Major cluster. 61 



Hinks, A. R., 240, 241 
Hough, S. S., 117, 124, 150 
Hyades, 56 

INNES, R. T. A., 18, 156 

Integral equations, 208, 219, 223 
"Island universe" theory, 242 

JKANS, J. H., 251 
Jones, H. S., 167 

KAPTEYN, J. C., distances of spectral 
types, 167 

fastest moving star, 18 

luminosity and density-laws, 216 

magnitude statistics, 191 

parallaxes, 41,60 

Perseus cluster. 64 

speeds of stars,' 154, 156 

star-streams, 87, 124, 164 

Type N stars, 182 
Kapteyn's mean parallaxes, 205 
Kapteyn's Plan of Selected Areas, 26 
King, E. S., 11 
Kobold, H., 83, 119, 138 
Kustner, F., 60 

LANE-RITTER Theory, 173 
Leavitt, Miss H. S. , 240 
Lockyer, Sir J. N., 10, 174 
Ludendorff, H., 10 
Luminosities of stars, 45 

dependence on spectral type, 48, 164 

of Taurus cluster, 60 
Luminosity law, 201 

Dyson and Eddington's results, 227 

Kapteyn's results, 216 

Seeliger's results, 214 

M TYPE stars, dual character of, 169 
Magellanic Clouds, 239 
Magnitudes of stars, 3 

absolute magnitudes, set Luminosities. 

stars of the two drifts, 109 

statistics, 188 
Masses of stars, 21, 160 
Maury, Miss A. C., 10, 175. 
Maxwellian law, 127, 151 
Melotte, P. J., 5, 186 
Milky Way, 196, 232 

bright stars in, 234 

distance of, 236 

equilibrium of, 260 

spiral theory, 244 
Monck, W. H., i:>4 
M t>t H.* /HI- it/ in rix, S(i 
Moving clusters, 54 

criteria for, 67 

dissolution of, !'.").'{ 

gravitation in, 'J .">:; 

iVi-rus duster, ")<i 

Taurus cluster, 64 

Ursa Major cluster, lil 



INDEX 



265 



N TYPE stars, 181 
Nebula?, dark, 237 

distribution of. -~ 

irregular, -'\~ 

planetary, 15<>, '24 1 

spiral, 242 

Xi-wi-omb, S., 76, 234 
Number of stars in the sky, 190, 195 

OBLATENKSS of stellar system, 200 
Observation, data of, "2 
Observational errors, effect on drift- 
constants, 147 

formula for correcting statistic^ _'_". i 
Orbits, stellar, time of describing, 255 
Orion, 63, 238 
Orion type of spectrum, 9, 159, 179 

PAHLKN, E. v. D., 242 
Parallactic motion, 79 

for different magnitudes, 203, 207 

for different types, 167 
Parallaxes, 12 ; accuracy of, 15 

Kapteyn's formulae, 205 

large (tables), 41, 47 

probable value, 222 

stars of the two drifts, 113, 116 

Taurus cluster, 60 
Parkhurst, J. A., 181 
Parsec, 14 

Periods of binary stars, 178 
Perseus cluster, 56 
Photographic magnitudes, 3 
Photometric magnitudes, 3 
Pickering, E. C., 8,165, 194 
Plan of Selected Areas, 26 
Planetary nebulae, 156, 241 
Pleiades, 63 
Poincare, H. , 260 
Precession, constant of, 73, 150 
Preferential motions, 87 
Proctor, R. A., 56 
Proper motions, accuracy of, 18 

analysis of, 88, 127 

cross motions, 158, 166 

large (list of), 19 

mean proper motion curves, 112, 116, 
139 

solar motion from, 73, 79 

statistical investigations of, 218 

RADIAL motions in stellar system, 257 
Radial Velocities, accuracy of, 20 

comparison with error-law, 151 

confirmation of star-streaming, 107 

ellipsoidal hypothesis, 145 

largest motions, 20 

progression with spectral type, 156 

solar motion from, 73, 84 

two-drift hypothesis, 143 
Regulus, 181 
Rigel, 45 
Rotation of 'Milky Way, 261 



Rudolph, K., 124 

Russell, H. X., 24, 17<>, 2m 

SAGITTARIUS region of Milky Wav, 38, 

197, 232, 258 
Schuster, A., 259 

Schwarzschikl, K., colour index, II 
effective wave length, 12 
ellipsoidal theory, 103, 124, 130 
luminosity law, 217 
magnitude statistics, 206 
parallaxes of spectral types, 167 
solution of statistical equations. _'( is 
Scorpius-Centaurus cluster, 70 
Scares, F. H., 5 

Secchi's classification of spectra, 10 
Seeliger, H., 213 
Selected Areas, plan of, 26 
Selection by magnitude, effect of, 46, 

172, 

Shapley, H., 24, 174 
Sirius, 22, 41, 50, 61, 63 
Sitter, W. de, 60 

Skewness of velocity-distribution, 135 
Solar motion, 71 ; dependence on spec- 
tral type, 77, 162 
direction of, 73, 102 
theory of determining, 79 
Spectral type, classification, 7 
evolution, 176 
in relation to density, 174 
galactic plane, 165, 169, 259 
luminosity, 48, 164 
mass, 160 
period, 178 
solar motion, 77, 162 
speed, 156, 259 
star-drifts, 110, 163 
Spectroscopic binaries, 20, 24, 44 
Speed, see Velocity. 
Spiral nebulae, 242 
Star-ratio, 184 
Star-streams, 86 

mathematical theory, 127 
suggested explanations, 244, 257 
tabulated results, 124 
Stellar system, dynamics of, 246 
evolution of, 259 
form and dimensions, 31 
oblateness of, 200 
Stumpe, O, 76 
Sun, motion in space, see Solar 

motion. 

position in stellar system, 31, 196 
stellar magnitude of, 6 

TABLES, 

(1) giving data for individual stars 
largest proper motions, 19 
masses (well-determined), 22 
motions, transverse and radial, 50 
parallaxes and luminosities, 41 , 47 



266 



INDEX 



Tables, Perseus cluster, 66 

stars with c-characteristic, 175 . 
Ursa Major cluster, rt3 

(2) <fii in;/ tteUisticnl rtxuff* 
colour index and >pectral type, 11 
densities of stars, 174 

distance distribution of stars, 216, 

221. 24, 226 
galactic distribution, 165, 181, 

239 

luminosities and spectra, 48 
luminosity-law, '2-27 
mean parallaxes, Kapteyn's, 20.") 

for spectra, 167 

velocities, for spectra, 156, 158 
number of stars in sky, 190 

(log B m ) 189, 190, 213 
periods of binaries, 178 
radial motions and error-law, 151, 

152 
solar motion, for magnitude, 76 

for spectra, 77, 162 
star-streams, magnitudes, 109 

mean parallaxes, 113 

number of stars, 117 

radial motions, 107 

spectra, 110 

(3) il/tiNtraf.iit.<i method* and theory 
comparison of hypotheses, 134 
star-stream analysis, 91, 116, 133, 

143 

(4) mathematical fmn-tions 
/(T), 129 
/)//(-. 132 

ff (T), 141 
(.")) .iii)imnry of ^tar-stream dfs- 

'># 

Api<-es of drifts, 125 
Vertex, 121 

Taurus clust.-r. 5ii, 253 
Thackeray, \V. <;.. 7-~> 
Three-drift hypothesis. 12i 
Turner, H. H., magnitude statistics, 

193 
parsec, 14 



Turner, H. H., proper motions, 19 
theory of star-streaming, 257 
Ursa Major system, 62 

Two-drift hypothesis, 90, 127 

comparison with ellipsoidal hypo- 
theses, 133, 22! l 
radial velocities, 143 

Two drifts, diagrams, 95 
distribution in the sky, 117 
division of stars between, 117 
in nearest stars, 53 
in stars of large proper motion, 1(>5 
magnitudes of stars in, 109 
mathematical theory, 127 
mean distances of, 111 
radial motions, 106 
spectra of stars in, 110 
suggested explanations, 244, 257 
summary of determinations, 124 

UNIT of distance, 14 

of drift- velocity (1 /h), 128 

of luminosity, 6 
Ursa Major cluster, 61 

VARIABLES, Cepheid, 6, 240 
Variables, Eclipsing, 24, 174, 239 
Velocities of stars, dependence on 

distance, 161 

dependence on type, 156, 259 
gravitational effects on, 246 
of intrinsically faint stars, 52 
of nearest stars. .">u 
origin of, 258 

Velocity-ellipsoid, 137, 145 
Velocity-law, 201, 220 
Vertex of star-streaming, 102 

determinations of, 124 
Visual magnitudes, 3, 193 

\V.\VK I.I:N<;THS, effective, 12 
Weersma, H A!, 41, 145, 103 
Whirlpool Nebula, 241 
Wolf, M., 238 
Wolf-Rayet stars, 8, 239 



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